Solvable real rigid Lie algebras are not necessarily completely solvable [Les alg`ebres de

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The Principal Element of a Frobenius Lie Algebra

A simple Lie algebra itself can never be Frobenius but many subalgebras are. For sln, where we may assume the simple roots numbered from 1 to n − 1, basic results of Elashvili [4], [2] assert in particular that the ith maximal parabolic subalgebra (obtained by deleting the ith negative root) is Frobenius if and only
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if i is prime to n. Dergachev and Kirillov [1] have given a simple algorithm to determine when a “seaweed” subalgebra of sln, i.e., one obtained by omitting both some positive and some negative roots, is Frobenius. The suggestive name comes from their shape as pictured in [1], but the deﬁnition is meaningful for all simple Lie algebras. Seaweed algebras are called “biparabolic” by A. Joseph [8].
The main results of this note are: 1) If a Frobenius Lie algebra f is a subalgebra of a simple Lie algebra g and if f is not an ideal of any larger subalgebra of g (a condition satisﬁed in particular when f contains a Cartan subalgebra of g) then the principal element is semisimple; we will call such a Frobenius subalgebra saturated. 2) If, moreover, g is sln then the eigenvalues of ad Fˆ are integers which, together with their multiplicities, are independent of the choice of Frobenius functional. In some important special cases where f has a “small” Frobenius functional, in particular for the seaweed subalgebras of sln, we give a simple method for the computation of the principal element.

Coalgebras of words and phrases

a r X i v :m a t h /0408258v 1 [m a t h .Q A ] 19 A u g 2004COALGEBRAS OF WORDS AND PHRASESVLADIMIR TURAEVAbstract.We introduce two constructions of a coassociative comultiplication in the algebra of phrases in agiven alphabet.As a preliminary step we give two constructions of a pre-Lie comultiplication in the modulegenerated by words.1.Introduction We give two new constructions of non-commutative non-cocommutative Hopf algebras (of inﬁnite rank).These constructions can be summarized as follows.A module M over a commutative ring gives rise to the tensor algebra W =T (M )=⊕m ≥0M ⊗m and to its positive subalgebra V =T +(M )=⊕m ≥1M ⊗m .Under certain additional assumptions,we deﬁne comultiplications in the tensor algebras P =T (V )and Q =T (W )so that they become Hopf algebras.A comultiplication ∆in a graded algebra ⊕m ≥0T m has a leading term which is the homomorphism T 1→T 1⊗T 1obtained by applying ∆and then projecting to T 1⊗T 1.This leading term yields the ﬁrst approximation to ∆and is often interesting in itself.The leading terms of our comultiplications in P and Q are homomorphisms V →V ⊗V and W →W ⊗W .They turn out to be pre-Lie comultiplications in the sense explained below.In particular,dualizing and skew-symmetrizing them,we obtain Lie brackets in the dual modules V ∗and W ∗.The study of tensor algebras can be reformulated in terms of words and phrases.Suppose from now on that the module M is free with basis A .Consider the basis of V =T +(M )formed by the vectors A 1⊗A 2⊗...⊗A m where m ≥1and A 1,A 2,...,A m ∈A .Omitting the symbol ⊗we can identify these vectors with words in the alphabet A .A basis in P =T (V )is formed by ﬁnite sequences of basis vectors in V ,that is by sequences of words.We call such sequences phrases .Natural bases in W =T (M )and Q =T (W )are similarly interpreted as words and phrases with the diﬀerence that here we allow an empty word.This interpretation gives a pleasant linguistical ﬂavour to the theory.It places the study of words and phrases in the setting of Hopf algebras,Lie algebras,and related algebraic objects.One well known construction of a comultiplication in W is based on the notion of a subword.For us,a word is a ﬁnite sequence of elements of A and a subword is any subsequence.Given a word w we can form the sum w ′⊗w/w ′where w ′runs over all subwords of w and the word w/w ′is obtained by deleting w ′from w .This yields the familiar shuﬄe comultiplication in W .It is coassociative and cocommutative.Our comultiplications are based on two diﬀerent constructions.They are determined by certain additionaldata which should be ﬁxed from the very beginning.The comultiplications in V and P =T (V )depend on a choice of a so-called stable set of words L in the alphabet A .We associate with a word w the sum w ′⊗w/w ′where w ′runs over all subwords of w formed by consecutive letters and belonging to L .This yields a pre-Lie comultiplication ρL in V .In a similar way we construct a coassociative comultiplication in P with leading term ρL .The second construction begins with ﬁxing a mapping µfrom A×A to the ground ring.The corresponding pre-Lie comultiplication ρµin W is deﬁned as follows.For a subword of length two a =AB of a word w ,set µa =µ(A,B )and denote by w ′a be the subword of w formed by the letters of w appearing between A and B .Let w ′′a be the word obtained by deleting both a and w ′a from w .Then ρµ(w )= a µa w ′a ⊗w ′′a .In a similar way we construct a coassociative comultiplication in Q =T (W )with leading term ρµ.Under a certain choice of µ,the latter comultiplication is closely related to the Connes-Kreimer comul-tiplication in the algebra of rooted trees,see [CK],or,more precisely,to its non-commutative version for planar rooted trees due to Foissy [Fo].Although we can directly deﬁne our comultiplications in P and Q ,we begin with a study of their leading terms.In Section 2we discuss relevant notions from the theory of pre-Lie multiplications and comultipli-cations.Sections 3and 4are concerned with the comultiplication in P derived from a stable set of words:12VLADIMIR TURAEVthe leading term is deﬁned in Section3and the comultiplication itself is deﬁned in Section4.Sections5 and6are concerned with the comultiplication in Q derived from a mappingµas above:the leading term is deﬁned in Section5and the comultiplication itself is deﬁned in Section6.At the end of Section6we discuss connections with the theory of planar rooted trees.Throughout the paper,weﬁx a commutative ring with unit R.The symbol⊗denotes the tensor product of R-modules over R.2.Pre-Lie coalgebras and left-handed bialgebras2.1.Pre-Lie algebras.By a multiplication in an R-module M we mean an R-bilinear mapping M×M→M. A(left)pre-Lie algebra over R is an R-module M endowed with a multiplication M×M→M,denoted⋆, such that for any f,g,h∈M,(2.1.1)f⋆(g⋆h)−(f⋆g)⋆h=g⋆(f⋆h)−(g⋆f)⋆h.The mapping⋆is called then a pre-Lie multiplication,see[Ge]and[Vi].For example,any associative multiplication is pre-Lie.A fundamental property of a pre-Lie multiplication⋆in M is that the formula [f,g]=f⋆g−g⋆f deﬁnes a Lie bracket in M.The Jacobi identity[f,[g,h]]+[g,[h,f]]+[h,[f,g]]=0for f,g,h∈M is a direct consequence of(2.1.1).In this way every pre-Lie algebra becomes a Lie algebra.There is a similar notion of right pre-Lie algebras.We will consider only left pre-Lie algebras and refer to them simply as pre-Lie algebras.Non-trivial examples of pre-Lie algebras can be obtained from derivations in algebras.A derivation in an associative R-algebra M is an R-linear homomorphism d:M→M such that d(fg)=d(f)g+f d(g)for any f,g∈M.An easy computation shows that for any a,b∈M,the formula f⋆g=afg+bfd(g)deﬁnes a pre-Lie multiplication in M.2.2.Pre-Lie coalgebras.A comultiplication in an R-module A is an R-linear homomorphismρ:A→A⊗A. We associate with suchρa homomorphism˜ρ:A→A⊗A⊗A sending a∈A to˜ρ(a)=(id A⊗ρ)ρ(a)−(ρ⊗id A)ρ(a).The comultiplicationρis coassociative if˜ρ=0.Dualizing Formula2.1.1,we obtain a notion of a pre-Lie ly,a pre-Lie coalgebra is an R-module A endowed with a comultiplicationρ:A→A⊗2=A⊗A such that for all a∈A,(2.2.1)P1,2˜ρ(a)=˜ρ(a).Here and below given a module U,we denote by P1,2the endomorphism of A⊗2⊗U permuting theﬁrst two tensor factors,i.e.,mapping a⊗b⊗u to b⊗a⊗u for a,b∈A,u∈U.A comultiplicationρsatisfying(2.2.1) is a pre-Lie comultiplication.Clearly,a coassociative comultiplication is pre-Lie.Given a pre-Lie coalgebra(A,ρ),consider the module A∗=Hom R(A,R)and the evaluation pairing , :A⊗A∗→R.The comultiplicationρinduces a pre-Lie multiplication⋆ρin A∗by(2.2.2) a,f⋆ρg = i a′i,f a′′i,g ∈Rfor any f,g∈A∗,a∈A and any(ﬁnite)expansionρ(a)= i a′i⊗a′′i∈A⊗2.In this way A∗becomes a pre-Lie algebra.odules over pre-Lie coalgebras.A(left)comodule over a pre-Lie coalgebra(A,ρ)is a pair(an R-module U,an R-homomorphismθ:U→A⊗U)such that˜θ=(id A⊗θ)θ−(ρ⊗id U)θ:U→A⊗2⊗U satisﬁes P1,2˜θ=˜θ.Given a comodule(U,θ)over(A,ρ),we deﬁne a right action of A∗on U byuf= i u′i,f u′′ifor any f∈A∗,u∈U,and any expansionθ(u)= i u′i⊗u′′i with u′i∈A,u′′i∈U.Lemma2.3.1.The action of A∗on U is a Lie algebra action.COALGEBRAS OF WORDS AND PHRASES3 Proof.We must prove that u[f,g]=(uf)g−(ug)f for all f,g∈A∗,u∈U where[f,g]=f⋆ρg−g⋆ρf.We shall use Sweedler’s notation for the expansionθ(u)= i u′i⊗u′′i and write it in the formθ(u)= (u)u′⊗u′′. Similarly,for a∈A,we writeρ(a)= (a)a′⊗a′′.By deﬁnition,(uf)g= (u) u′,f u′′g= (u),(u′′) u′,f (u′′)′,g (u′′)′′.Hence(uf)g−(ug)f= (u),(u′′) u′,f (u′′)′,g (u′′)′′− (u′′)′,f u′,g (u′′)′′.To compute u[f,g],observe that for a∈A,a,[f,g] = a,f⋆ρg−g⋆ρf = (a) a′,f a′′,g − a′′,f a′,g .Thereforeu[f,g]= (u) u′,[f,g] u′′= (u),(u′) (u′)′,f (u′)′′,g u′′− (u′)′′,f (u′)′,g u′′.The equality u[f,g]=(uf)g−(ug)f follows now from the formula(u′)′⊗(u′)′′⊗u′′−(u′)′′⊗(u′)′⊗u′′= (u),(u′′)u′⊗(u′′)′⊗(u′′)′′−(u′′)′⊗u′⊗(u′′)′′(u),(u′)which is a reformulation of the equality P1,2˜θ=˜θ.Lemma2.3.1implies that the formula fu=−uf with u∈U,f∈A∗deﬁnes a left Lie algebra action of A∗on U.It is clear that(A,θ=ρ:A→A⊗A)is a comodule over(A,ρ).Lemma2.3.1yields a Lie algebra action of A∗on A.2.4.Left-handed bialgebras.By a bialgebra we shall mean a pair(T,∆)where T is an associative unital R-algebra and∆:T→T⊗T is a coassociative algebra comultiplication in T.The words“algebra comulti-plication”mean that∆(1)=1⊗1and∆(ab)=∆(a)∆(b)for any a,b∈T.Here multiplication in T⊗T is deﬁned by(a⊗a′)(b⊗b′)=ab⊗a′b′for a,b,a′,b′∈T.We do not require the existence of a counit T→R although in all our constructions of bialgebras there will be a counit.The comultiplication in a bialgebra(T,∆)induces an associative multiplication in T∗=Hom R(T,R) by fg(a)= (a)f(a′)g(a′′)for a∈T,f,g∈T∗,and any expansion∆(a)= (a)a′⊗a′′.This makes T∗into an associative algebra.If T has a counit,then T∗is a unital algebra.Dualizing multiplication in T we obtain a homomorphism from T∗to a certain completion of T∗⊗T∗.We call this homomorphism quasi-comultiplication.A bialgebra(T,∆)is graded if T splits as a direct sum of submodules T=⊕m≥0T m such that1∈T0and T m T n⊂T m+n for all m,n.Set T+=⊕m≥1T m⊂T.A graded bialgebra(T,∆)is left-handed if for any a∈T1,∆(a)−a⊗1−1⊗a∈T+⊗T1.The leading term of∆is then the homomorphism(π⊗π)∆|T1:T1→T1⊗T1whereπ:T→T1is the projection.We shall mainly apply these deﬁnitions in the case whereT=T(A)=⊕m≥0A⊗mis the tensor algebra of an R-module A.Here A⊗0=R,A⊗1=A,and A⊗m is the tensor product of m copies of A for m≥2.The next lemma relates left-handed bialgebras with pre-Lie coalgebras.Lemma 2.4.1.Let(T=⊕m≥0T m,∆)be a left-handed graded bialgebra such that T1generates T+=⊕m≥1T m.Then the leading termρ:T1→T1⊗T1of∆is a pre-Lie comultiplication in T1.Proof.For a∈T1,we can expand∆(a)=a⊗1+1⊗a+ i a′i⊗a′′i+ j a1j a2j⊗a3j(mod⊕m≥3T m⊗T1)4VLADIMIR TURAEVwhere i,j run overﬁnite sets of indices and a′i,a′′i,a1j,a2j,a3j∈T1.Clearly,ρ(a)= i a′i⊗a′′i.Our assump-tions imply that∆(T m)⊂⊕k+n≥m T k⊗T n for all puting(id⊗∆)∆(a)and(∆⊗id)∆(a)modulo ⊕k,n,r,k+n+r≥4T k⊗T n⊗T r and equating the resulting expressions we obtain that˜ρ(a)= j a1j⊗a2j⊗a3j+a2j⊗a1j⊗a3j.Therefore P1,2˜ρ(a)=˜ρ(a).Let(T,∆)be as in the lemma.A(left)comodule over(T,∆)is a pair(an R-module U,an R-homomorphism Θ:U→T⊗U)such that(∆⊗id U)Θ=(id T⊗Θ)ΘandΘ(u)−1⊗u∈T+⊗U for any u∈U.Then U becomes a right module over the algebra T∗by uf= (u)f(u′)u′′for f∈T∗,u∈U,and anyﬁnite expansionΘ(u)= (u)u′⊗u′′.For example,set U=T1and deﬁneΘ:U→T⊗U byΘ(u)=∆(u)−u⊗1 for u∈U.The pair(U,Θ)is a comodule over(T,∆)(cf.the proof of Theorem4.2.1below).This makes U=T1into a right T∗-module.2.5.Remarks.1.Lemma2.4.1suggests to study when a given pre-Lie comultiplication is induced froma left-handed comultiplication in a graded bialgebra.For the pre-Lie comultiplications deﬁned in the next sections this will be always the case.2.Consider a pre-Lie coalgebra(A,ρ)and a submodule B⊂A such thatρ(A)⊂B⊗A andρ(B)⊂B⊗B. Then(B,ρ|B)is a pre-Lie coalgebra and(A,θ=ρ:A→B⊗A)is a comodule over(B,ρ|B).The associated action of B∗on A is compatible with the action of A∗on A via the Lie algebra homomorphism A∗→B∗induced by the inclusion B⊂A.3.For a pre-Lie coalgebra(A,ρ),the homomorphismρ−P1,2ρ:A→A⊗2is a Lie cobracket(cf.,for instance[Tu]).In particular in Lemma2.4.1,ρ−P1,2ρis a Lie cobracket in T1.This still holds if the left-handedness assumption on∆is weakened to∆(a)−a⊗1−1⊗a∈T+⊗T+for all a∈T1.4.Lemma2.4.1extends to comodules as follows.Let(T,∆)be as in this lemma.The leading term ofa comodule(U,Θ)over(T,∆)is the homomorphismθ=(π⊗id)Θ:U→T1⊗U whereπ:T→T1is the projection.A direct computation shows that(U,θ)is a comodule over the pre-Lie coalgebra(T1,ρ= (π⊗π)∆|T1).Note also that if T0=R,then the projection T→T0=R is a counit of(T,∆).3.Coalgebra of words3.1.Words.For us,an alphabet is an arbitrary set and letters are its elements.Throughout the paper we ﬁx an alphabet A.A word of length m≥1is a mapping from the set{1,2,...,m}to A.By deﬁnition,there is a unique word of length0called the empty word and denotedφ.To present a word w:{1,2,...,m}→A it is enough to write down the sequence of letters w(1)w(2)...w(m).For instance,the symbol ABA represents the word{1,2,3}→A sending1and3to A∈A and sending2to B∈A.Writing down consecutively the letters of two words w and x we obtain their concatenation wx.For instance,the concatenation of w=ABA and x=BB is the word wx=ABABB.A word x is a factor of a word w if w=yxz for certain words y,z.A factor x of w is proper if x=w.Given a word w of length m≥1and numbers1≤i≤j≤m+1,set w i,j=w(i)w(i+1)...w(j−1). This word of length j−i is a factor of w=w1,i w i,j w j,m+1.The word w i,j is empty iﬀi=j and proper iﬀ(i,j)=(1,m+1).Let W=W(A)be the free R-module freely generated by the set of words in the alphabet A.A typical element of W is aﬁnite formal linear combination of words with coeﬃcients in R.Each word w represents a vector in W denoted also w.These vectors form a basis of W.Let V=V(A)be the submodule of W generated by non-empty words.Clearly,W=V⊕Rφ.3.2.Stable sets of words.A set L of non-empty words is stable if it satisﬁes the following condition:(∗)For any word w of length m≥1and any indices1≤i<j≤m+1such that(i,j)=(1,m+1)and w i,j∈L,the word w belongs to L if and only if the word w1,i w j,m+1belongs to L.The words belonging to a stable set L will be called L-words.The“only if”part of(∗)means that striking out from any L-word a proper factor belonging to L one obtains an L-word.The“if”part of(∗)means that inserting an L-word in an L-word one obtains an L-word.In particular,concatenation of L-words is an L-word.We give a few examples of stable sets of words.(1)The set of all non-empty words and the void set of words are stable.COALGEBRAS OF WORDS AND PHRASES5 To give the next example,denote by f A(w)the number of appearances of a letter A∈A in a word w.For instance,f A(ABA)=2.(2)Pick A∈A.The set of non-empty words w such that f A(w)=0is stable.Similarly,for any integer N,the set of non-empty words w such that f A(w)is divisible by N is stable.(3)Pick letters A1,...,A n∈A and elements g1,...,g n of a certain abelian group.The set of non-emptywords w such that f A1(w)g1+...+f An(w)g n=0is stable.Observeﬁnally that the intersection of any family of stable sets of words is stable.A union of stable sets can be non-stable.3.3.Pre-Lie coalgebra of words.Weﬁx a stable set of words L and derive from it a pre-Lie comultipli-cationρL in the module V.A simple cut of a word w of length m≥1is a pair of indices1≤i<j≤m+1such that(i,j)=(1,m+1) and w i,j∈L.To indicate that(i,j)is a simple cut of w we write(i,j)≺w.SetρL(w)= (i,j)≺w w i,j⊗w1,i w j,m+1∈V⊗Vwhere the sum runs over all simple cuts of w.Note that the words w i,j and w1,i w j,m+1are necessarily non-empty.This deﬁnesρL on the basis of V and extends by linearity to a comultiplicationρL:V→V⊗V. For example,if A,B∈A and L is the set of all non-empty words,thenρL(A)=0,ρL(AB)=A⊗B+B⊗A,ρL(ABA)=A⊗BA+B⊗AA+A⊗AB+AB⊗A+BA⊗A.Theorem3.3.1.ρL is a pre-Lie comultiplication in V.Proof.For a word w of length m≥1,(id⊗ρL)ρL(w)= (i,j)≺w w i,j⊗ρL(w1,i w j,m+1).To compute w i,j⊗ρL(w1,i w j,m+1),we need to consider factors of w1,i w j,m+1.There are three kinds of them: (i)the factors of w1,i−1,(ii)the factors of w j+1,m+1,and(iii)the factors obtained by concatenation of some w i′,i with some w j,j′where i′≤i,j≤j′.Here and below i,j,i′,j′∈{1,2,...,m}.The factors of w1,i w j,m+1 of type(i)contribute to w i,j⊗ρL(w1,i w j,m+1)the expressionx i,j=(i′,j′)≺w,j′<iw i,j⊗w i′,j′⊗w1,i′w j′,i w j,m+1.The factors of w1,i w j,m+1of type(ii)contributey i,j=(i′,j′)≺w,j<i′w i,j⊗w i′,j′⊗w1,i w j,i′w j′,m+1.The factors of w1,i w j,m+1of type(iii)contributez i,j=i′≤i,j≤j′,(i′,j′)=(1,m+1),w i′,i w j,j′∈Lw i,j⊗w i′,i w j,j′⊗w1,i′w j′,m+1.Recall that w i,j∈L.By deﬁnition of a stable set of words,w i′,i w j,j′∈L iﬀ(i′,j′)=(i,j)and w i′,j′∈L. Therefore the conditions on i′,j′in the latter sum are equivalent to the conditions i′≤i,j≤j′,(i′,j′)≺w,(i′,j′)=(i,j).We obtain(i,j)≺w z i,j=(i,j)≺w,(i′,j′)≺w,i′≤i,j≤j′,(i′,j′)=(i,j)w i,j⊗w i′,i w j,j′⊗w1,i′w j′,m+1 = (i′,j′)≺wρL(w i′,j′)⊗w1,i′w j′,m+1=(ρL⊗id)ρL(w).Hence,˜ρL(w)=(id⊗ρL)ρL(w)−(ρL⊗id)ρL(w)= (i,j)≺w x i,j+y i,j.It remains to observe thatP1,2( (i,j)≺w x i,j)= (i,j)≺w y i,j.6VLADIMIR TURAEVTherefore˜ρL(w)is invariant under P1,2.3.4.Word indicators.The elements of the module V∗=Hom R(V,R)are called word indicators.The module V∗admits a decreasingﬁltration V∗=V∗(1)⊃V∗(2)⊃...where V∗(m)consists of the word indicators annihilating all non-empty words of length≤m−1.Clearly,∩m V∗(m)=0.We can consider inﬁnite sums in V∗as follows.Let f1,f2,...∈V∗be a sequence of word indicators such that for any m≥1all terms of the sequence starting from a certain place belong to V∗(m).Then for any a∈V,the sum f(a)=f1(a)+f2(a)+···contains only aﬁnite number of non-zero terms and the formula a→f(a):V→R deﬁnes a word indicator f=f1+f2+···.A similar construction shows that the natural homomorphism V∗→proj lim m(V∗/V∗(m)) is an isomorphism.By the general theory,the pre-Lie comultiplicationρL in V induces a pre-Lie multiplication⋆L and a Lie bracket[f,g]L=f⋆L g−g⋆L f in V∗.It is clear that V∗(m)⋆L V∗(n)⊂V∗(m+n)and[V∗(m),V∗(n)]L⊂V∗(m+n) for all m,n.Therefore V∗=proj lim m(V∗/V∗(m))is a projective limit of nilpotent Lie algebras.Recall the Lie algebra action of V∗on V induced byρL.For f∈V∗and a word w of length m≥1,fw=−wf=− (i,j)≺w w i,j,f w1,i w j,m+1.All word indicators annihilating L lie in the kernel of this action.The module V has an increasingﬁltration0=V0⊂V1⊂V2⊂...where V m is generated by non-empty words of length≤m.It is clear that f V m⊂V m−1for all m and f∈V∗.Thus the action of V∗on V m is nilpotent for all m.The restriction ofρL:V→V⊗V to the module RL⊂V generated by L is a pre-Lie comultiplication in RL sinceρL(RL)⊂RL⊗RL.ClearlyρL(V)⊂RL⊗V so that V becomes a comodule over the pre-Lie coalgebra RL.The associated actions of(RL)∗and V∗on V are compatible via the Lie algebra homomorphism V∗→(RL)∗induced by the inclusion RL⊂V.This is a special case of Remark2.5.2.3.5.Example.Let the alphabet A consist of only one letter A.Denote by A m the word consisting of m letters A where m≥1.A stable set of words L is determined by a positive integer N and consists of all words A m with m divisible by N.The comultiplicationρL is computed byρL(A m)= k≥1,kN<m(m+1−kN)A kN⊗A m−kNfor any m≥1.We can identify a word indicator f with its generating function m≥1f(A m)t m.One easily computes that f⋆L g=(f|N)·(g+tg′)where for a formal power series f= m≥1a m t m,we set f|N= m≥1a mN t mN and f′= m≥1ma m t m−1.For N=1,we obtain f⋆L g=fg+tfg′.3.6.The group Exp L V∗.Assume in this subsection that R⊃Q.For f,g∈V∗,setfg=f+g+112([f,[f,g]L]L+[g,[g,f]L]L)+ (V)where the right-hand side is the Campbell-Baker-Hausdorﬀseries for log(e f e g).The resulting mapping V∗×V∗→V∗is a group multiplication in V∗.Here f−1=−f and0∈V∗is the group unit.The resulting group is denoted Exp L V∗.Heuristically,this is the“Lie group”with Lie algebra(V∗,[,]L).The equality V∗=proj lim m(V∗/V∗(m))implies that Exp L V∗is pro-nilpotent.The action of V∗on V induced byρL integrates to a group action of Exp L V∗on V.To see this,denote byϕ(f)the additive endomorphism a→fa of V determined by f∈V∗.Seteϕ(f)(a)= k≥01COALGEBRAS OF WORDS AND PHRASES7 3.7.Remarks.1.There is an embeddingδ:V֒→V∗mapping a word w into the word indicatorδw whose value on w is1and whose value on all other words is0.It is easy to check that the image ofδis closed under⋆L.This induces a pre-Lie multiplication◦L in V.For words w=A1A2...A m and x=B1B2...B n with m,n≥1,we have w◦L x=0if w/∈L andw◦L x=δ−1(δw⋆Lδx)=ni=0B1...B i A1...A m B i+1...B nif w∈L.In the case where L is the set of all non-empty words,this pre-Lie multiplication in V is essentially due to Gerstenhaber[Ge].2.With a slight modiﬁcation,the notion of a stable set of words can be used to deﬁne a coassociative comultiplication in W=V⊕Rφ.Let us say that a set of words S is strongly stable ifφ∈S and for any word w and any its subword w′we have w′∈S⇒(w∈S⇔w/w′∈S)where w/w′is the word obtained by deleting w′from w.Examples of strongly stable sets of words can be obtained by adjoining the empty word to the stable sets of words in the examples in Section3.2.For a strongly stable set S and a word w, set∆S(w)= w′w′⊗w/w′where w′runs over all subwords of w belonging to S.It is easy to see that∆S extends to a coassociative algebra comultiplication in W.When S is the set of all words,this is the shuﬄe comultiplication mentioned in the introduction.4.Hopf algebra of phrasesThe results above suggest that there may exist a left-handed comultiplication in the tensor algebra T(V) with leading termρL.We construct such a comultiplication.4.1.Phrases.A phrase of length k≥1is a sequence of k words in the alphabet A.Some(or all)of these words may be empty.We also allow an empty phrase consisting of0words and denoted1.A more interesting example of a phrase:“NIHIL NOV I SUB SOLE”where A is the set of capital Latin letters.Here words are separated by blank spaces and the quotation marks indicate the beginning and the end of the phrase. These conventions,customary in ordinary texts,are not quite convenient for mathematical formulas.In the formulas we shall indicate the beginning and the end of a phrase by round brackets and separate words by vertical bars.Of course,we assume that the round brackets and the vertical bar are not letters in A.The same example can be re-written as(NIHIL|NOV I|SUB|SOLE).By abuse of notation,the phrase(w) consisting of one word w will be also denoted by w.Let Q=Q(A)be the free R-module freely generated by the set of phrases in the alphabet A.Concatenation of phrases deﬁnes a multiplication in Q by(w1|w2|...|w k)(x1|x2|...|x t)=(w1|w2|...|w k|x1|x2|...|x t)where k,t≥0and w1,...,w k,x1,...,x t are words.This makes Q into an associative algebra with unit1(the empty phrase).This algebra is graded,the grading being the length of phrases.In this section we shall focus on the subalgebra P=P(A)of Q additively generated by phrases formed by non-empty words.Thus(w1|w2|...|w k)∈P iﬀk=0or k≥1and all the words w1,...,w k are non-empty. Note that1∈P.The algebra P is graded:P=P0⊕P1⊕...where P k is the module additively generated by phrases of length k.The inclusion V=P1֒→P extends to an algebra isomorphism T(V)→P and we identify the tensor algebra T(V)with P along this isomorphism.Phrases in the alphabet A can be viewed as words in the extended alphabet A∐{|}obtained by adjoining the new letter|to A.However,multiplication of phrases is diﬀerent from multiplication of words in this extended alphabet due to the additional symbol|between w k and x1in the formula above.ultiplication in P.Fix a stable set of words L.We deﬁne here a comultiplication∆=∆L in P. We begin with preliminary deﬁnitions.Let w be a word of length m≥1.A cut of w of length k≥0is a sequence of2k integers1≤i1<j1<i2<j2<...<i k<j k≤m+1such that the k words w i1,j1,...,w ik,j kbelong to L.For k=1,we additionallyrequire that(i1,j1)=(1,m+1).A cut of length1is nothing but a simple cut in terminology of Section3.3. Every word has a unique empty cut∅of length0.8VLADIMIR TURAEVTo indicate that c=(i1,j1,...,i k,j k)is a cut of w we write c≪w.We also write#c={1,2,...,k}.Foru∈#c,set w c u=w iu ,j u.The factors w c1,...,w c k of w are called c-factors.Finally we deﬁne a phrase l c(w)= u∈#c(w c u)=(w c1|w c2|...|w c k)and a wordr c(w)=w1,i1w j1,i2w j2,i3...w jk−1,i kw jk,m+1.To specify a cut of w of length k it is enough to specify k non-overlapping proper factors of w belonging to L and such that consecutive factors are separated by at least one letter.These factors form the phrase l c(w). Deleting them from w we obtain the word r c(w).Set∆(w)=w⊗1+ c≪w l c(w)⊗r c(w)where c runs over all cuts of w.Note that the term corresponding to the empty cut is1⊗w.The mapping w→∆(w)extends uniquely to an algebra homomorphism∆=∆L:P→P⊗2.For example,for a word w of length1,we have∆(w)=w⊗1+1⊗w.If L is the set of all non-empty words and A,B,C∈A,then∆(AB)=AB⊗1+1⊗AB+A⊗B+B⊗A,∆(ABC)=ABC⊗1+1⊗ABC+A⊗BC+B⊗AC+C⊗AB+AB⊗C+BC⊗A+(A|C)⊗B,∆(AB|C)=∆(AB)∆(C)=(AB|C)⊗1+C⊗AB+(A|C)⊗B+(B|C)⊗A+AB⊗C+1⊗(AB|C)+A⊗(B|C)+B⊗(A|C).Theorem4.2.1.The pair(P,∆)is a left-handed graded bialgebra with leading termρL.Proof.The left-handedness and the claim concerning the leading term follow directly from the deﬁnitions.The only non-obvious assertion is the coassociativity of∆.It suﬃces to prove that(id⊗∆)∆(w)=(∆⊗id)∆(w) for any word w.SetΘ(w)=∆(w)−w⊗1= c≪w l c(w)⊗r c(w).The mapping w→Θ(w)deﬁnes an R-linear homomorphismΘ:V→P⊗V.We have(∆⊗id)∆(w)=(∆⊗id)Θ(w)+∆(w)⊗1=(∆⊗id)Θ(w)+Θ(w)⊗1+w⊗1⊗1. Similarly,(id⊗∆)∆(w)=(id⊗∆)Θ(w)+(id⊗∆)(w⊗1)=(id⊗Θ)Θ(w)+Θ(w)⊗1+w⊗1⊗1. Comparing these expressions we conclude that it is enough to prove that(∆⊗id)Θ(w)=(id⊗Θ)Θ(w).If follows from the deﬁnitions that(id⊗Θ)Θ(w)= c≪w e≪r c(w)l c(w)⊗l e(r c(w))⊗r e(r c(w)).We compute the right-hand side as follows.For integers i<j,set[i,j)={s∈Z|i≤s<j}.For cuts d=(i1,j1,...,i k,j k),d′=(i′1,j′1,...,i′k′,j′k′)of w,write d⊂d′if∪k u=1[i u,j u)⊂∪k′v=1[i′v,j′v).Suppose that d⊂d′.We say that the index v∈{1,...,k′}is special if i′v=i u and j′v=j u for some u=1,...,k.If v is non-special,then clearly[i′v,j′v) ∪k u=1[i u,j u). We can obtain a cut˜d′of r d(w)by deleting all d-factors from the d′-factors of w numerated by non-special indices.That the˜d′-factors of r d(w)belong to L follows from the stability of L.The cut˜d′is empty iﬀd=d′. Moreover,the formula(d,d′)→(d,˜d′)establishes a bijective correspondence between pairs(d≪w,d′≪w) with d⊂d′and pairs(c≪w,e≪r c(w)).Therefore(id⊗Θ)Θ(w)=d≪w,d′≪w,d⊂d′l d(w)⊗l˜d′(r d(w))⊗r˜d′(r d(w))=d≪w,d′≪w,d⊂d′l d(w)⊗l˜d′(r d(w))⊗r d′(w).We claim that the right-hand side is equal to(∆⊗id)Θ(w).(The proof of this does not use the stability of L).If follows from the deﬁnitions that(∆⊗id)Θ(w)= c≪w u∈#c w c u⊗1+ e u≪w c u l e u(w c u)⊗r e u(w c u) ⊗r c(w)COALGEBRAS OF WORDS AND PHRASES9= c≪w I⊂#c{e u≪w c u}u∈#c−Iv∈Iw c v u∈#c−I l e u(w c u) ⊗ u∈#c−I r e u(w c u)⊗r c(w).Here all the products are ordered in accordance with the natural order in#c.For example,if#c={1,2,3} and I={1,3},then the term in the big round brackets on the right-hand side is w c1l e2(w c2)w c3.Given a cut c=(i1,j1,...,i k,j k)of w and u∈#c,every cut e u of w c u yields a cutˆe u of w by adding i u−1 to all terms of e u.A set I⊂#c gives rise to a cutˆI of w formed by the indices{i v,j v}v∈I.With a tuple (c≪w,I⊂#c,{e u≪w c u}u∈#c−I)we associate a pair(d,d′)where d′=c and d is the cut of w obtained as the union ofˆI with all{ˆe u}u∈#c−I.This establishes a bijective correspondence between such tuples and the pairs(d≪w,d′≪w)with d⊂d′.The corresponding terms in the expansions of(id⊗Θ)Θ(w)and (∆⊗id)Θ(w)are equal.Therefore(id⊗Θ)Θ(w)=(∆⊗id)Θ(w). Corollary4.2.2.The bialgebra(P,∆)is a Hopf algebra.Proof.The augmentationε:P→R sending all non-empty phrases to0and sending1to1is a counit of∆. It remains to show the existence of an antipode.Consider the unique R-linear endomorphism s of P such that s(1)=1,s(ab)=s(b)s(a)for all a,b∈P and the value of s on words is deﬁned by induction on the length as follows:for a word w of length1,set s(w)=−w;for a word w of length≥2,sets(w)=−w− c≪w,c=∅l c(w)s(r c(w))∈Pwhere we use that r c(w)is shorter than w.These formulas imply thatµ(id⊗s)∆(w)=ε(w)whereµis multiplication in P.In other words,s is a right inverse of id:P→P with respect to the(associative) convolution product•in End R(P)deﬁned by f•g=µ(f⊗g)∆for f,g∈End R(P).Similar inductive formulas show that id has a left inverse s′∈End R(P)and then s′=s′•(id•s)=(s′•id)•s=s.Therefore s is an antipode for(P,∆).4.3.Phrase indicators.By phrase indicators we mean R-linear homomorphisms P→R.By the general theory of bialgebras,the comultiplication∆L in P induces an associative multiplication◦L in the module of phrase indicators P∗=Hom R(P,R).The product f◦L g may distinguish phrases indistinguishable by f,g∈P∗.For example,let L consist of all non-empty words and letℓand f B be the phrase indicators counting the number of words in a phrase and the number of appearances of B∈A in a phrase,respectively. Then the values ofℓ◦L f B on the1-word phrases(ABC)and(ACB)(where A,B,C are distinct letters in A)are4and3,respectively.The additive homomorphismΘ:V→P⊗V constructed in the proof of Theorem4.2.1makes V into a comodule over the bialgebra(P,∆).This induces a right action of the algebra P∗on ing the antiautomorphism of P∗induced by the antipode in P,we can transform the right action of P∗into a left action.The right and left actions of a phrase indicator f∈P∗on V depend only on the values of f on phrases with all words in L.This can be formalized as follows.Let P L be the subalgebra of P additively generated by the phrases whose all words belong to L(including the empty phrase1).It is clear that∆L(P L)⊂P L⊗P L so that P L is a Hopf subalgebra of P.Clearly,Θ(V)⊂P L⊗V.In this way V acquires the structure of a comodule over P L.The actions of the algebras P∗,(P L)∗on P are compatible via the algebra homomorphism P∗→(P L)∗induced by the inclusion P L⊂P.4.4.Dual Hopf algebra.We can deﬁne a Hopf algebra dual to P.Consider the algebra P∗with multipli-cation induced by∆L and quasi-comultiplication induced by multiplication in P.Consider the embedding δ:P֒→P∗mapping a phrase p into the phrase indicatorδp whose value on p is1and whose value on all other phrases is0.It is easy to see thatδ(P)is a subalgebra of P∗.In this way the module P acquires a new associative multiplication◦L.The quasi-comultiplication in P∗induces a genuine comultiplication in P transforming a phrase(w1|...|w k)into k i=0(w1|...|w i)⊗(w i+1|...|w k).This makes the algebra(P,◦L) into a Hopf algebra.By its very deﬁnition,it is dual to(P,∆L).For completeness,we describe multiplication◦L in P explicitly.For a phrase p=(w1|...|w k)and a non-empty word y,set p∗y=0if at least one of the words w1,...,w k does not belong to L.If w1,...,w k∈L, setp∗y= y=x1···x k+1x1w1x2w2...w k x k+1∈V⊂P。

LEIBNIZ n-ALGEBRAS

1. Introduction
Leibniz algebras were introduced by the second author in [4]. They play an important role in Hochschild homology theory [4], [5] as well as in Nambu mechanics ([6], see also [1]). Let us recall that a Leibniz algebra is a vector space g equipped with a bilinear map [−, −] : g ⊗ g → g satisfying the identity : (1.1) [x, [y, z ]] = [[x, y ], z ] − [[x, z ], y ].
2. Derivations
In the whole paper K is a ﬁeld. All tensor products are taken over K . Let A be a vector space equipped with an n-linear operation ω : A⊗n → A. A map f : A → A is called a derivation with respect to ω if
LEIBNIZ n-ALGEBRAS
J. M. CASAS,(1) J.-L. LODAY
(1) (2)
and T. PIRASHVILI(3)
(2)
Dpto Matematica Aplicada, Universidad de Vigo, 36005 Pontevedra, Spain IRMA, ULP et CNRS, 7, rue R. Descartes, 67084 Strasbourg, France (3) Math. Inst., Alexidze str. 1, 380093 Tbilisi, Republic of Georgia

Generalized Drinfeld realization of quantum superalgebras and $U_q(hat {frak osp}(1,2))$

$Generalized Drinfeld realization of quantum superalgebras and $U_q(hat {frak osp}(1,2))$$

a r X i v :m a t h /99587v3[mat h.QA ]13Mar21GENERALIZED DRINFELD REALIZATION OF QUANTUM SUPERALGEBRAS AND U q (ˆosp (1,2))JINTAI DING AND BORIS FEIGIN Dedicated to our friend Moshe Flato Abstract.In this paper,we extend the generalization of Drin-feld realization of quantum aﬃne algebras to quantum aﬃne su-peralgebras with its Drinfeld comultiplication and its Hopf algebra structure,which depends on a function g (z )satisfying the relation:g (z )=g (z −1)−1.In particular,we present the Drinfeld realization of U q (ˆosp (1,2))and its Serre relations.1.Introduction.Quantum groups as a noncommutative and noncocommutative Hopf algebras were discovered by Drinfeld[Dr1]and Jimbo[J1].The standard deﬁnition of a quantum group is given as a deformation of universal enveloping algebra of a simple (super-)Lie algebra by the basic genera-tors and the relations based on the data coming from the correspond-ing Cartan matrix.However,for the case of quantum aﬃne algebras,there is a diﬀerent aspect of the theory,namely their loop realizations.The ﬁrst approach was given by Faddeev,Reshetikhin and Takhtajan [FRT]and Reshetikhin and Semenov-Tian-Shansky [RS],who obtained a realization of the quantum loop algebra U q (g ⊗C [t,t −])via a canon-ical solution of the Yang-Baxter equation depending on a parameter z ∈C .On the other hand,Drinfeld [Dr2]gave another realization ofthe quantum aﬃne algebra U q (ˆg )and its special degeneration called the Yangian,which is widely used in constructions of special representation of aﬃne quantum algebras[FJ].In [Dr2],Drinfeld only gave the real-ization of the quantum aﬃne algebras as an algebra,and as an algebra this realization is equivalent to the approachabove [DF]through cer-tain Gauss decomposition for the case of U q (ˆgl (n )).Certainly,the mostimportant aspect of the structures of the quantum groups is its Hopf algebra structure,especially its comultiplication.Drinfeld also con-structed a new Hopf algebra structure for this loop realization.The new comultiplication in this formulation,which we call the Drinfeld1comultiplication,is simple and has very important applications[DM] [DI2].In[DI],we observe that in the Drinfeld realization of quantum aﬃne algebras U q(ˆsl(n)),the structure constants are certain rational func-tions g ij(z),whose functional property of g ij(z)decides completely the Hopf algebra structure.In particular,for the case of U q(ˆsl2),its Drin-feld realization is given completely in terms of a function g(z),which has the following function property:g(z)=g(z−1)−1.This leads us to generalize this type of Hopf ly,we can substitute g ij(z)by other functions that satisfy the functional property of g ij(z),to derive new Hopf algebras.In this paper,we will further extend the generalization of the Drinfeld realization of U q(ˆsl2)to derive quantum aﬃne superalgebras.As an ex-ample,we will also present the quantum aﬃne superalgebra U q(ˆosp(1,2)) in terms of the new formulation,in particular,we present the Serre re-lations in terms of the current operators.The paper is organized as the following:in Section2,we recall the main results in[DI]about the generalization of Drinfeld realization of U q(ˆsl(2));in Section3,we present the deﬁnition of the generalized Drinfeld realization of quantum superalgebras;in Section4,we present the formulation of U q(ˆosp(1,2)).2.In[DI],we derive a generalization of Drinfeld realization of U q(ˆsl n). For the case of U q(sl2),weﬁrst present the complete deﬁnition.Let g(z)be an analytic functions satisfying the following property that g(z)=g(z−1)−1andδ(z)be the distribution with support at1. Deﬁnition2.1.U q(g,f sl2)is an associative algebra with unit1and the generators:x±(z),ϕ(z),ψ(z),a central element c and a nonzero complex parameter q,where z∈C∗.ϕ(z)andψ(z)are invertible.In2terms of the generating functions:the deﬁning relations are ϕ(z)ϕ(w)=ϕ(w)ϕ(z),ψ(z)ψ(w)=ψ(w)ψ(z),g(z/wq−c)ϕ(z)ψ(w)ϕ(z)−1ψ(w)−1=c)±1x±(w),2ψ(z)x±(w)ψ(z)−1=g(w/zq∓1δ(z2c)−δ(z2c) ,q−q−1x±(z)x±(w)=g(z/w)±1x±(w)x±(z).Theorem2.1.The algebra U q(g,f sl2)has a Hopf algebra structure, which are given by the following formulae.Coproduct∆(0)∆(q c)=q c⊗q c,(1)∆(x+(z))=x+(z)⊗1+ϕ(zq c12),(3)∆(ϕ(z))=ϕ(zq−c22),(4)∆(ψ(z))=ψ(zq c22),where c1=c⊗1and c2=1⊗c.Counitεε(q c)=1ε(ϕ(z))=ε(ψ(z))=1,ε(x±(z))=0.Antipode a(0)a(q c)=q−c,(1)a(x+(z))=−ϕ(zq−c2)−1,(3)a(ϕ(z))=ϕ(z)−1,(4)a(ψ(z))=ψ(z)−1.Strictly speaking,U q(g,f sl2)is not an algebra.This concept,which we call a functional algebra,has already been used before[S],etc.3The Drinfeld realization for the case of U q(ˆsl2)[Dr2]as a Hopf algebra is diﬀerent,and it an algebra and Hopf algebra deﬁned with current operators in terms of formal power series.Let g(z)be an analytic functions that satisfying the following prop-erty that g(z)=g(z−1)−1=G+(z)/G−(z),where G±(z)is an analytic function without poles except at0or∞and G±(z)have no common zero point.Letδ(z)= n∈Z z n,where z is a formal variable.Deﬁnition2.2.The algebra U q(g,sl2)is an associative algebra with unit1and the generators:¯a(l),¯b(l),x±(l),for l∈Z and a central element c.Let z be a formal variable andx±(z)= l∈Z x±(l)z−l,ϕ(z)= m∈Zϕ(m)z−m=exp[ m∈Z≤0¯a(m)z−m]exp[ m∈Z>0¯a(m)z−m] andψ(z)= m∈Zψ(m)z−m=exp[ m∈Z≤0¯b(m)z−m]exp[ m∈Z>0¯b(m)z−m]. In terms of the formal variables z,w,the deﬁning relations are a(l)a(m)=a(m)a(l),b(l)b(m)=b(m)b(l),g(z/wq−c)ϕ(z)ψ(w)ϕ(z)−1ψ(w)−1=c)±1x±(w),2ψ(z)x±(w)ψ(z)−1=g(w/zq∓1δ(z2c)−δ(z2c) ,q−q−1G∓(z/w)x±(z)x±(w)=G±(z/w)x±(w)x±(z),where by g(z)we mean the Laurent expansion of g(z)in a region r1> |z|>r2.Theorem2.2.The algebra U q(g,sl2)has a Hopf algebra structure. The formulas for the coproduct∆,the counitεand the antipode a are the same as given in Theorem2.1.Here,one has to be careful with the expansion of the structure func-tions g(z)andδ(z),for the reason that the relations between x±(z) and x±(z)are diﬀerent from the case of the functional algebra above.4Example2.1.Let¯g(z)be a an analytic function such that¯g(z−1)=−z−1¯g(z).Let g(z)=q−2¯g(q2z)g(z/wq c),ϕ(z)x±(w)ϕ(z)−1=g(z/wq∓12c)∓1x±(w),{x+(z),x−(w)}=1wq−c)ψ(wq1wq c)ϕ(zq1Accordingly we have that,for the tensor algebra,the multiplication is deﬁned for homogeneous elements a,b,c,d by(a⊗b)(c⊗d)=(−1)[b][c](ac⊗bd),where[a]∈Z2denotes the grading of the element a.Similarly we have:Theorem3.1.The algebra U q(g,f s)has a graded Hopf algebra struc-ture,whose coproduct,counit and antipode are given by the same for-mulae of U q(q,f sl2)in Theorem2.1.As for the case of U q(g,f sl2)is not a graded algebra but rather a a graded functional algebra.Letg(z)=g(z−1)−1=G+(z)/G−(z),where G±(z)is an analytic function without poles except at0or∞and G±(z)have no common zero point.Deﬁnition3.2.The algebra U q(g,s)is Z2graded associative algebra with unit1and the generators:¯a(l),¯b(l),x±(l),for l∈Z and a central element c,where x±(l)are graded1(mod2)and the rest are graded o(mod2).Let z be a formal variable andx±(z)= l∈Z x±(l)z−l,ϕ(z)= m∈Zϕ(m)z−m=exp[ m∈Z≤0¯a(m)z−m]exp[ m∈Z>0¯a(m)z−m] andψ(z)= m∈Zψ(m)z−m=exp[ m∈Z≤0¯b(m)z−m]exp[ m∈Z>0¯b(m)z−m]. In terms of the formal variables z,w,the deﬁning relations are ϕ(z)ϕ(w)=ϕ(w)ϕ(z),ψ(z)ψ(w)=ψ(w)ψ(z),g(z/wq−c)ϕ(z)ψ(w)ϕ(z)−1ψ(w)−1=c)±1x±(w),2ψ(z)x±(w)ψ(z)−1=g(w/zq∓1δ(z2c)−δ(z2c) ,q−q−1(G∓(z/w))x±(z)x±(w)=−(G±(z/w))x±(w)x±(z),6where by g(z)we mean the Laurent expansion of g(z)in a region r1> |z|>r2.The above relations are basically the same as in that of Deﬁnition2.2 except the relation between x±(z)and x±(w)respectively,which diﬀers by a negative sign.The expansion direction of the structure functions g(z)andδ(z)is very important,for the reason that the relations be-tween x±(z)and x±(z)are diﬀerent from the case of the functional algebra above.Theorem3.2.The algebra U q(g,s)has a Hopf algebra structure.The formulas for the coproduct∆,the counitεand the antipode a are the same as given in Theorem2.1.Example3.1.Let¯g(z)= 1.From[CJWW][Z],we can see that U q(1,s)is basically the same as U q(ˆgl(1,1)).4.For a rational function g(z)that satisﬁesg(z)=g(z−1)−1,it is clear that g(z)is determined by its poles and its zeros,which are paired to satisfy the relations above.For the simplest case(except g(z)=1)that g(z)has only one pole and one zero,we havezp−1g(z)=zp2−1z−p1zp2−1z−p1As in[DM][DK],for the case of quantum aﬃne algebras,it is very important to understand the poles and zero of the product of current operators.We will start with the relations between X+(z)with itself.From the deﬁnition,we know that(z−p1w)(z−p2w)X+(z)X+(w)=−(zp1−w)(zp2−w)X+(w)X+(z).From this,we know that X+(z)X+(w)has two poles,which arelocated at(z−p1w)=0and(z−p2w)=0.This also implies thatProposition4.1.X+(z)X+(w)=0,when z=w.If we assume that U q(g,s)is related to some quantized aﬃne super-algebra,then we can see that the best chance we have is U q(ˆosp(1,2))by looking at the number of zeros and poles of X+(z)X+(w).However,for the case of U q(ˆosp(1,2)),we know we need an extraSerre relation.For this,we will follow the idea in[FO].Let Letf(z1,z2)=(z1−p1z2)(z1−p2z2).(z−p1w)(z−p2w)Y+(z,w)=(z1−p1z3)(z1−p2z3)z1−z2X+(z1)X+(z2)X+(z3),z2−z3F(z1,z2,z3)=(z1−p1z2)(z1−p2z2)(z3−p1z1)(z3−p2z1)(z2−p1z3)(z2−p2z3)=f(z1,z2)f(z2,z3)f(z3,z1),¯F(z,z2,z3)=f(z2,z1)f(z2,z3)f(z3,z1).1Let V(z1,z2,z3)be the algebraic variety of the zeros of F(z1,z2,z3).Let V(a(z1),a(z2),a(z3)),be the image of the action of a on this variety,where a∈S3,the permutation group on z1,z2,z3.Let¯V(z1,z2,z3)bethe algebraic variety of the zeros of¯F(z1,z2,z3).Let¯V(a(z1),a(z2),a(z3)),be the image of the action of a on this variety,where a∈S3.Proposition4.2.Y+(z,w)has no poles and is symmetry with respectto z and w.Y+(z1,z2,z3)has no poles and is symmetry with respectto z1,z2,z3.8Following the idea in[FO],we would like to deﬁne the following conditions that may be imposed on our algebra.Zero Condition I:Y+(z1,z2,z3)is zero on at least one line that crosses(0,0,0),and this line must lie in a V(a(z1),a(z2),a(z3))for some element a∈S3Zero Condition II:Y+(z1,z2,z3)is zero on at least one line that crosses(0,0,0),and this line must lie in a¯V(a(z1),a(z2),a(z3))for some element a∈S3For the line that crosses(0,0,0)where Y+(z1,z2,z3)is zero,we call it the zero line of Y+(z1,z2,z3).Because Y+(z1,z2,z3)is symmetric with respect to the action of S3on z1,z2,z3,if a line is the zero line of Y+(z1,z2,z3),then clearly the orbit of the line under the action of S3 is also an zero line.Remark1.There is a simple symmetry that we would prefer to choose the function F(z1,z2,z3)to determine the variety V(z1,z2,z3). We have thatF(z1,z2,z3)=f(z1,z2)f(z2,z3)f(z3,z1).Let S12be the permutation group acting on z1,z2.Let S22be the permu-tation group acting on z2,z3.Let S12be the permutation group acting on z3,z1.Clearly,[FO]we can choose from a family of varieties determined by the functions f(a1(z1),a1(z2))f(a2(z2),a2(z3))f(a3(z3),a3(z1))for a1∈S12,a2∈S22,a3∈S32.For each such a functionf(a1(z1),a1(z2))f(a2(z2),a2(z3))f(a3(z3),a3(z1)),we can attach a oriented diagram,whose nods are z1,z2,z3,and the arrows are given by(a1(z1)→a1(z2)),(a2(z2)→a2(z3))(a3(z3)→a3(z1)).For example the diagram of F(z1,z2,z3)is given byCase1Let q2=p−11,which implies that p−21must be either p1or p2. Clearly,it can not be p1,which implies that the impossible condition p1=1Therefore,we have thatp−21=p2,which is what we want.Case2Let q2=p−12,which implies that p1p2must be either p1or p2.Clearly,it can not be p1because it implies p2=1,it can not be neither be p2,which implies that p1=1.This completes the proof forp2=p−21.Similarly,if we have that q1=p2,we can,then,showp1=p−22.However from the algebraic point of view,the two condition are equivalent in the sense that p1and p2are symmetric.Also we have thatProposition4.5.If we impose the Zero condition II on the algebra U q(g,s),we havep1=p22,orp2=p21.However we also have that:If we impose the Zero condition II on the algebra U q(g,s), then U q(g,s)is not a Hopf algebra anymore.The reason is that the Zero condition II can not be satisﬁed by comultiplication,which can be checked by direct calculation.This is the most important reason that we will choose the Zero con-dition I to be imposed on the algebra U q(g,s),which comes actually from the consideration of Hopf algebra ly,if we choose V(z1,z2,z3)or the equivalent ones which has the same diagram presen-tation as Diagram I to deﬁne the zero line of Y+(z1,z2,z3),then,the quotient algebra derived from the Zero Condition I is still a Hopf algebra with the same Hopf algebra structure(comultiplication,counit and antipode).11From now on,we impose the Zero condition I on the algebra U q(g,s),and let usﬁx the notation such thatp1=q2,p2=q−1.Similarly,we deﬁne(z−p−11w)(z−p−12w)Y−(z,w)=(z1−p−11z3)(z1−p−12z3)z1−z2X+(z1)X+(z2)X+(z3),z2−z3We now deﬁne the q-Serre relation.q-Serre relationsY+(z1,z2,z3)is zero on the linez1=z2q−1=z3q−2.Y−(z1,z2,z3)is zero on the linez1=z2q=z3q2.The q-Serre relations can also be formulated in more algebraic way.Proposition4.6.The q-Serre relations are equivalent to the followingtwo relations:(z3−z1q−1)(z3−z1q3)(z1−z2q2)X+(z3)X+(z2)X+(z1))−(z1−z2q−1)(z3−z1q)((z1−z3q2)(z1−z3q)(z1q−z3q−1)(z1−z2q2)X+(z2)X+(z1)X+(z3))=0, (z1−z3)(z3−z1q2)(z1−z2q−1)(z3−z1q)(z3−z1q−3)(z1−z2q−2)(z2−z1q−2)(z2−z1q)(z3−z1q)(z3−z1q−3)X−(z1)X−(z2)X−(z3)−(z1−z3)(z3−z1q−2)((z1−z3q−2)(z1−z3q−1)(z1q−z3q)(z2−z1q−2)(z2−z1q)[DI]J.Ding,K.Iohara Generalization and deformation of the quantum aﬃne algebras Lett.Math.Phys.,41,1997,181-193q-alg/9608002,RIMS-1091 [DI2]J.Ding,K.Iohara Drinfeld comultiplication and vertex operators,Jour.Geom.Phys.,23,1-13(1997)[DK]J.Ding,S.Khoroshkin Weyl group extension of quantized current algebras, to appear in Transformation Groups,QA/9804140(1998)[DM]J.Ding and T.Miwa Zeros and poles of quantum current operators and the condition of quantum integrability,Publications of RIMS,33,277-284(1997)[Dr1]V.G.Drinfeld Hopf algebra and the quantum Yang-Baxter Equation,Dokl.Akad.Nauk.SSSR,283,1985,1060-1064[Dr2]V.G.Drinfeld Quantum Groups,ICM Proceedings,New York,Berkeley, 1986,798-820[Dr3]V.G.Drinfeld New realization of Yangian and quantum aﬃne algebra, Soviet Math.Doklady,36,1988,212-216[Er] B.Enriquez,On correlation functions of Drinfeld currents and shuﬄe al-gebras,math.QA/9809036.[FRT]L.D.Faddeev,N.Yu,Reshetikhin,L.A.Takhtajan Quantization of Lie groups and Lie algebras,Yang-Baxter equation in Integrable Systems,(Ad-vanced Series in Mathematical Physics10)World Scientiﬁc,1989,299-309. [FO] B.Feigin,V.Odesski Vector bundles on Elliptic curve and Sklyanin alge-bras RIMS-1032,q-alg/9509021[FJ]I.B.Frenkel,N.Jing Vertex representations of quantum aﬃne algebras, A85(1988),9373-9377[GZ]M.Gould,Y.Zhang On Super RS algebra and Drinfeld Realization of Quantum Aﬃne Superalgebras q-alg/9712011[J1]M.Jimbo A q-diﬀerence analogue of U(g)and Yang-Baxter equation,Lett.Math.Phys.10,1985,63-69[RS]N.Yu.Reshetikhin,M.A.Semenov-Tian-Shansky Central Extensions of Quantum Current Groups,LMP,19,1990[S] E.K.Sklyanin On some algebraic structures related to the Yang-Baxter equation Funkts.Anal.Prilozhen,16,No.4,1982,22-34[Z]Y.Zhang Comments on Drinfeld Realization of Quantum Aﬃne Superal-gebra U q[gl(m|n)(1)]and its Hopf Algebra Structure q-alg/9703020 Jintai Ding,Department of Mathematical Sciences,University of CincinnatiBoris Feigin,Landau Institute of Theoretical Physics14。

The Spectral Geometry of the Equatorial Podles Sphere

a rX iv:mat h /4834v2[mat h.QA ]27Ma y25The Spectral Geometry of the Equatorial Podle´s Sphere Ludwik D¸a browski 1,Giovanni Landi 2,Mario Paschke 3,Andrzej Sitarz 4∗1Scuola Internazionale Superiore di Studi Avanzati,Via Beirut 2-4,I-34014,Trieste,Italy 2Dipartimento Matematica e Informatica,Universita di Trieste via A.Valerio 12/1,I-34127,Trieste,Italy 3Max-Planck-Institut f¨u r Mathematik in den Naturwissenschaften,Inselstr.22,04103Leipzig,Germany 4Institute of Physics,Jagiellonian University,Reymonta 4,30-059Krak´o w,Poland Abstract We propose a slight modiﬁcation of the properties of a spectral geometry a la Connes,allowing for some of the algebraic relations to be satisﬁed modulo compact operators.On the equatorial Podle´s sphere we construct U q (su (2))-equivariantDirac operator and real structure which satisfy these modiﬁed properties.Key words and phrases :Noncommutative geometry,spectral triple,quantum SU (2).Mathematics Subject Classiﬁcation:Primary 58B34;Secondary 17B37.1IntroductionRecent examples of noncommutative spectral geometries on spaces coming from quantum groups[2,7,4,14,11,8]opened a new interesting and promising research area.Along these lines,introducing a slight modiﬁcation of the deﬁning properties of a noncommuta-tive geometry,we present a spectral geometry for the equatorial Podle´s sphere S2q of[13]. Both the Dirac operator D and the real structure J are equivariant under the action of U q(su(2))on S2q.With q a real number,0<q≤1,we denote by A(S2q)the algebra of polynomial func-tions generated by operators a,a∗and b=b∗,which satisfy the following commutation rules:ba=q2ab,a∗b=q2ba∗,a∗a+b2=1,q2aa∗+q−2b2=q2.This algebra has an‘equator’-worth of classical points given by the one dimensional representations b=0,a=λ,withλ∈S1.The symmetry of the sphere,which we shall use for the equivariance,is the Hopf algebra module structure with respect to the U q(su(2))Hopf algebra.It is derived from the canonical U q(su(2))action on the A(SU q(2))algebra.Explicitly,the generators e,f,k of U q(su(2))act on the generators of A(S2q)in the following way:k⊲a=qa,e⊲a=−(1+q2)q−5a∗,e⊲a∗=0,f⊲a∗=(1+q2)q−3q2a∗,f⊲b=−q3The essentially unique Dirac operator,which comes out as the solution of the above condition,shall have the crucial property that all commutators[D,π(x)],x∈A(S2q),are bounded.3The equivariant geometry of A(S2q)A fully equivariant approach,both for the real structure and the Dirac operator was worked out for the standard Podle´s sphere in[7].Here we build up on this and an earlier approach[9](see also[12])to construct an equivariant spectral geometry of the Equatorial Podle´s sphere.The starting ingredient of the equivariant spectral geometry,which we are about to construct is an equivariant representation of A(S2q)on a Hilbert space H.Let us recall that an H-equivariant representation of an H-module algebra A on H,is a representation πof A on H and a representationρof H on a dense subspace of H such that for every a∈A and h∈H and v from a dense subspace we have:ρ(h)π(a)v=π(h(1)⊲a)ρ(h(2))v.(4) In our case,this is the same as a representation of the crossed product of U q(su(2))⋉A(S2q) on a dense subspace of the Hilbert space.The general theory for the entire family of Po-dle´s spheres is in[15].Proposition3.1.There exists two irreducible U q(su(2))-equivariant representations(de-noted byπ±)of the algebra A(S2q)on the Hilbert space H h= l=12,...V l given by:π±(a)|l,m =±(1+q2)q m−1[2l][2l+2]2[l+m+1][l+m+2]|l+1,m+1−q m+l+1[2l][2l][2l+2][l−m+1][l+m]−q2[l−m][l+m+1] |l,m−q m+1[l−m+1][l+m+1]|l+1,m−q m+1[l−m][l+m]|l−1,m ,(5)withπ(a∗)being the hermitian conjugate ofπ(a)and[x]:=(q−q−1)−1(q x−q−x).The proof is a long but straightforward calculation based on the covariance prop-erty(4)with the natural representationρof U q(su(2))on H h and the U q(su(2))-module structure of A(S2q)given in(1).The representationsπ±are equivalent to the left regular3representation of A(SU q(2))on L2(SU q(2))(with the Haar measure)when this represen-tation is restricted to the subalgebra A(S2q)and the representation space is restricted to the(L2-completion)of certain vector spaces(left A(S2q)-modules)constructed in[1].We take as the Hilbert space of our geometry H=H h⊕H h,with the natural grading γ=id⊕(−id)and the representationπ(x)=π+(x)⊕π−(x)for any a∈A(S2q),which is equivariant with respect to the diagonal action(which we call againρ)of U q(su(2))on H.Like in[7]we follow the method of equivariance toﬁndﬁrst the real structure J.Let us recall,that J is the antiunitary part of an antilinear operator T on H,which must then satisfy for any h∈U q(su(2))(on a dense subspace of H),ρ(h)T=Tρ(Sh)∗.Taking into account the required commutation relations with the gradingγ,that isγJ=−Jγand J2=−1,one easily obtain that J must be,J|l,m ±=i2m|l,−m ∓,(6) where the label±refers to the two copies of H h which are marked by the eigenvalues of γ.With this data we immediately meet an obstruction:Proposition3.2.The operator J deﬁned above does not satisfy the“commutant”re-quirement of a real spectral triple,that is,there exist x∈A(S2q)for which Jπ(x)J−1is not in the commutant ofπ(A(S2q)).We move on to look for a variation of spectral geometry up to inﬁnitesimal.Let K denotes the ideal of compact operators on H and K q⊂K be the ideal generated by operators L q of the form L q|l,m ±=q l|l,m ±.Using compact perturbations of the representationπone proves the followingProposition 3.3.The operator J,deﬁned in(6),mapsπ(A(S2q ))to its commutantmodulo compact operators(in fact modulo K q).More precisely,for any x,y∈A(S2q),π(x),Jπ(y)J−1 ∈K q.(7) Exact formulæshall be contained in the extended version of this note.As a next step we derive the Dirac operator D.Beside postulating that D anticom-mutes withγand commutes with J,we shall also require that D is equivariant,that is it commutes with the representationρof U q(su(2))on H.Each operator satisfying these condition must be of the formD|l,m ±=d±l |l,m ∓,d±l∈R.(8)Proposition3.4.Up to rescaling and addition of an odd constant as well as of an odd element from K q,there exists only one operator D of the form(8)which satisﬁes the order-one condition up to compact operators(in fact modulo K q),that is for all x,y∈A(S2q),Jπ(x)J−1,[D,π(y)] ∈K q.(9)For this operator D,the parameters d±l are given by d+l=d−l=l+1The condition(9)has been verify explicitly for all pairs of generators of A(S2q)with the help of a symbolic computation program.Furthermore,we have,),the commutators[D,π(x)]are bounded. Proposition3.5.For any x∈A(S2qIt is evident that the operator D is self-adjoint on a natural domain in H and that its resolvent is compact.Since the spectrum of|D|consists only of eigenvalues k=l+1[12]M.Paschke,¨Uber nichtkommutative Geometrien,ihre Symmetrien und etwas Hoch-energiephysik,Thesis,Mainz2001.[13]P.Podle´s,Quantum spheres,Lett.Math.Phys.14(1987)521–531.[14]K.Schm¨u dgen,E.Wagner,Dirac operator and a twisted cyclic cocycle on the stan-dard Podle´s quantum sphere,math.QA/0305051.[15]K.Schm¨u dgen,E.Wagner,Representations of cross product algebras of Podle´s quan-tum spheres,math.QA/0305309.[16]A.Sitarz,Equivariant spectral triples,Banach Center Publ.61(2003)231–263.6。

invariants and K-spectrums of local theta lifts

arXiv:1302.1031v2 [math.n 1.1. Let WR be a ﬁnite dimensional symplectic real vector space. Throughout this paper (G, G′) will denote a type I irreducible reductive dual pair in Sp(WR ). Such dual pairs are listed in Table 1 in Section 2.1. We follow the notation in [15] closely. Let Sp(WR ) be the metaplectic double cover of Sp(WR ). For any subgroup E of Sp(WR ), let E denote its inverse image in Sp(WR ). We choose a maximal compact subgroup U of Sp(WR ) such that K := G ∩ U and K ′ := G′ ∩ U are maximal compact subgroups of G and G′ respectively. Hence K and K ′ are maximal compact subgroups of G and G′ respectively. The choice of U determines a unique complex structure on WR with the following property: there is a positive deﬁnite Hermitian form , on the resulting complex vector space W so that the imaginary part of , coincides with the symplectic form on WR , and U coincides with the unitary group attached to (W, , ). We choose the oscillator representation of Sp(WR ) whose Fock model Y is realized as the space C [W ] of complex polynomials on W with the U action as described in Appendix A.1. Let ς denote the minimal U-type of Y . It is a one dimensional representation of U acting on the space of constant functions in C [W ]. Let ς |E denote the restriction of ς to E for any subgroup E of U. Let g = k ⊕ p denote the complexiﬁed Cartan decomposition of the Lie algebra of G corresponding to the maximal compact subgroup K . Likewise we deﬁne g′ = k′ ⊕ p′ for G′ . Let ρ be an irreducible admissible genuine (g, K )-module. By (2.5) in [15], Y /(∩ψ∈Homg,K (Y ,ρ) ker ψ ) ≃ ρ ⊗ Θ(ρ) where Θ(ρ) is a (g′ , K ′ )-module called the full (local) theta lift of ρ. Theorem 2.1 in [15] states that if Θ(ρ) = 0, then Θ(ρ) is a (g′ , K ′ )-module of ﬁnite length with an inﬁnitesimal character and it has a unique irreducible quotient θ(ρ) called the (local) theta lift of ρ.

Lie algebroid structures and Lagrangian systems on affine bundles

a r X i v :m a t h /023178v1[mat h.DG ]18Mar22Lie algebroid structures and Lagrangian systems on aﬃne bundles Eduardo Mart´ınez †,Tom Mestdag ‡and Willy Sarlet ‡†Departamento de Matem´a tica Aplicada Universidad de Zaragoza,Mar ´ıa de Luna 3,50015Zaragoza,Spain ‡Department of Mathematical Physics and Astronomy Ghent University,Krijgslaan 281,B-9000Ghent,Belgium Abstract As a continuation of previous papers,we study the concept of a Lie algebroid structure on an aﬃne bundle by means of the canonical immer-sion of the aﬃne bundle into its bidual.We pay particular attention to the prolongation and various lifting procedures,and to the geometrical construction of Lagrangian-type dynamics on an aﬃne Lie algebroid.1Introduction Since the book of Mackenzie [12],the mathematics of Lie algebroids (and groupoids)has been studied by many authors;for a non-exhaustive list of references,see for example [3,5,6,9,10,13,18].The potential relevance of Lie algebroids for applications in physics and other ﬁelds of applied mathematics has gradu-ally become more evident.In particular,contributions by Libermann [11]and Weinstein [21]have revealed the role Lie algebroids play in modelling certain problems in mechanics.The concept of ‘Lagrangian equations’on Lie algebroids certainly deﬁnes an interesting generalisation of Lagrangian systems as known from classical mechanics,if only because of the more general class of diﬀerentialequations it involves while preserving a great deal of the very rich geometrical structure of Lagrangian (and Hamiltonian)mechanics.One of us [16],in particular,has produced evidence of this rich structure by showing that one can prolong a Lie algebroid in such a way that the newly obtained space has all the features of tangent bundle geometry,which are im-portant for the geometrical construction of Lagrangian systems.That is to say,the prolonged Lie algebroid carries a Liouville-type section and a vertical endomorphism which enables the deﬁnition of a Poincar´e -Cartan type 1-form,associated to a function L ;the available exterior derivative then does the rest1for arriving at an analogue of the symplectic structure from which Lagrangian equations can be derived.In[20],we have started an investigation on the possible generalisation of the concept of Lie algebroids to aﬃne bundles.Our principle motivation was to create a geometrical model which would be a natural environment for a time-dependent version of Lagrange equations on Lie algebroids,as discussed for example in[21]and[16].Since classical time-dependent mechanics is usually described on theﬁrst-jet bundle J1M of a manifold Mﬁbred over I R(see e.g. [4,14]),and J1M→M is an aﬃne bundle,it looks natural to build up a time-dependent generalisation in such a way that J1M→M is the image bundle of the anchor map of a Lie algebroid structure on some aﬃne bundle E→M. An additional indication that such a set-up is well suited came from a naive calculus of variations approach,which gives a clue on the analytical format one should expect for such time-dependent Lagrange equations(see[19]).Lie algebroids on vector bundles are known to give rise,among other things, to a linear Poisson structure on the dual bundle,as well as a coboundary opera-tor on its Grassmann algebra;in fact,these properties equivalently characterise the Lie algebroid structure.One of the features of the approach to aﬃne al-gebroids adopted in[20]was our speciﬁc choice to develop,in a direct way, a consistent theory of forms on sections of an aﬃne bundle and their exterior calculus.By contrast,however,in the context of brieﬂy mentioning the related Poisson structure in the concluding remarks,we did announce a forthcoming diﬀerent approach,which would be based on the fact that an aﬃne bundle can be regarded as an aﬃne sub-bundle of a vector bundle,namely the dual of its extended dual.This is the line of reasoning we will develop here;it could be termed‘indirect’because it makes use of an imbedding into a larger bundle, but it has some marked advantages,such as the fact that proving a number of properties becomes much easier and that new insights come to the forefront.As a matter of fact,one readily recognises via this approach that much(if not all) of the theory of aﬃne Lie algebroids can be developed without needing an extra ﬁbration of the base manifold over I R.We will accordingly start our present analysis in this more general set-up and brieﬂy come back to the special case appropriate for time-dependent systems in the concluding remarks.Note:while the very last editing of this paper was being done,we have been informed of similar investigations on aﬃne algebroids,which have been carried out by Grabowski et al[7].The reader mayﬁnd it instructive to compare the two simultaneous developments,which are related to each other up to and including our Section6.The scheme of the paper is as follows.A fairly detailed description of purely algebraic aspects of the theory is given in Sections2to4;it involves the introduc-tion of the concept of a Lie algebra over an aﬃne space and aspects of exterior calculus.A Lie algebroid structure on a general aﬃne bundleτ:E→M is deﬁned in Section5:essentially,it comes from a classical Lie algebroid on the dual˜E of the extended dual E†of E,with the property that the bracket of sections in the image of the inclusion i:E→˜E,lies in the image of the under-lying vector bundle.Equivalent characterisations of this property can be found2in the subsequent section on the exterior diﬀerential and the associated Poisson structure.Section7presents a number of simple examples of aﬃne algebroids. The important concept of prolongation of an algebroid is discussed in Section8: starting from a general construction on vector algebroids,it is shown that the prolonged bundle inherits the aﬃne structure coming from E when the vector algebroid is the one on˜E.For the speciﬁc case of interest,it is further shown (see Section9)that there is a canonical map which gives rise to a‘vertical en-domorphism’on sections of the prolonged bundle.Natural constructions which are then available are complete and vertical lifts;they play a role in the ge-ometric deﬁnition of Lagrangian systems on aﬃne Lie algebroids,presented in Section10.2Immersion of an aﬃne space in a vector space Let A be an aﬃne space modelled on a vector space V,and let A†=Aﬀ(A,I R) be the extended dual of A,that is,the vector space of all aﬃne maps from A to the real line.We consider the bidual˜A of A,in the sense˜A=(A†)∗.It is well known that in the case of a vector space V,the bidual˜V=(V∗)∗is isomorphic to V.In the case of an aﬃne space,the bidual includes‘a copy’of A,as it is shown in the following statement.Proposition1.The map i:A→˜A given by i(a)(ϕ)=ϕ(a)is an injective aﬃne map,whose associated vector map is i:V→˜A given by i(v)(ϕ)=ϕ(v) Proof.If a∈A and v∈V,then for allϕ∈A†,i(a+v)(ϕ)=ϕ(a+v)=ϕ(a)+ϕ(v)=i(a)(ϕ)+i(v),from which it follows that i is an aﬃne map whose associated linear map is i.To prove that i is injective,it suﬃces to prove that i is injective,which is obvious since if v is an element in the kernel of i then i(v)(ϕ)=ϕ(v)=0for allϕ∈V∗,hence v=0.The vector space˜A is foliated by hyperplanes parallel to the image of i. Every vector z∈˜A is either of the form z=i(v)for some v∈V or of the form z=λi(a)for someλ∈I R\{0}and a∈A.Moreover,λ,a and v are uniquely determined by z.The image of the map i consists of the points for whichλ=1. To understand this description in more detail,we will prove that we have an exact sequence of vector spaces0−→V i−→˜A−→I R−→0.To this end we consider the dual sequence.Proposition2.Let l:I R→A†be the map that associates toλ∈I R the con-stant functionλon A.Let k:A†→V∗the map that associates to every aﬃne function on A the corresponding linear function on V.Then,the sequence of vector spaces0−→I R l−→A†k−→V∗−→0is exact.3Proof.Indeed,it is clear that l is injective,k is surjective and k◦l=0,so that Im(l)⊂Ker(k).Ifϕ∈A†is in the kernel of k,that is the linear part ofϕvanishes,then for every pair of points a and b=a+v we have thatϕ(b)=ϕ(a+v)=ϕ(a)+ϕ(v)=ϕ(a),that isϕis constant,and hence in the image of l.The dual map of k is i,since for v∈V we havek(ϕ),v = ϕ,v = ϕ,i(v)The dual map j of the map l is given by j(αi(a)+iv)=α.Indeed,for every λ∈I R we havej(z)λ= z,l(λ) = αi(a)+i(v),l(λ) =α i(a)),l(λ) + i(v),l(λ) =αλIt follows thatCorollary1.If A isﬁnite dimensional,then the sequence0−→V i−→˜A j−→I R−→0is exact.Note that in this way we can clearly identify the image of V as the hyperplane of˜A with equation j(z)=0,and the image of A as the hyperplane of˜A with equation j(z)=1,in other wordsi(V)=j−1(0)and i(V)=j−1(1).Note in passing that if we have an exact sequence0−→Vα−→W j−→I R−→0, then we can deﬁne A=j−1(1);it follows that A is an aﬃne space modelled on the vector space V and W is canonically isomorphic to˜A.The isomorphism is the dual map ofΨ:W∗→A†,Ψ(φ)(a)=φ(i(a)),where i:A→W is the canonical inclusion.We now discuss the construction of a basis for A†.Let(O,{e i})be an aﬃne frame on A.Thus every point a has a representation a=O+v i e i.The family of aﬃne maps{e0,e1,...,e n}given bye0(a)=1e i(a)=v i,is a basis for A†.Ifϕ∈A†,and we putϕ0=ϕ(O)andϕi=ϕ(e i),then ϕ=ϕ0e0+ϕi e i.It is to be noticed that,contrary to e1,...,e n,the map e0does not depend on the frame we have chosen for A.In fact,e0coincides with the map j.Let now{e0,e1,...,e n}denote the basis of˜A dual to{e0,e1,...,e n}.Then the image of the canonical immersion is given byi(O)=e0i(e i)=e i4from which it follows that for a=O+v i e i,we have i(a)=e0+v i e i.If we denote by(x0,x1,...,x n)the coordinate system on˜A associated to the basis {e0,...,e n},then the equation of the image of the map i is x0=1,while the equation of the image of i is x0=0.Coordinates in˜A∗=A†associated to the basis above will be denoted by (µ0,µ1,...,µn),that isµα(ϕ)= eα,ϕ for everyϕ∈A†.3Lie algebra structure over an aﬃne space Implicit in our previous paper[20]is the following deﬁnition of a Lie algebra over an aﬃne space.Deﬁnition1.Let A be an aﬃne space over a vector space V.A Lie algebra structure on A is given by•a Lie algebra structure[,]on V,and•an action by derivations of A on V,i.e.a map D:A×V→V,(a,v)→D a v with the propertiesD a(λv)=λD a vD a(v+w)=D a v+D a wD a[v,w]=[D a v,w]+[v,D a w],•satisfying the compatibility propertyD a+v w=D a w+[v,w].Incidentally,it is suﬃcient to require in theﬁrst item that the bracket on V is I R-bilinear and skew-symmetric,since the Jacobi identity then follows from the requirements on D a.If we use a bracket notation[a,v]≡D a v,then the conditions in the deﬁni-tion above read[a,λv]=λ[a,v][a,v+w]=[a,v]+[a,w][a,[v,w]]=[[a,v],w]+[v,[a,w]][a+v,w]=[a,w]+[v,w].This allows us to deﬁne a bracket of elements of A by putting[v,a]=−[a,v]and,if b=a+v,then[a,b]=[a,v].This bracket is skew-symmetric by construction and also satisﬁes a Jacobi-type property.5Theorem1.A Lie algebra structure over an aﬃne space A is equivalent to a Lie algebra extension of the trivial Lie algebra I R by V.Explicitly,it is equivalent to the exact sequence of vector spaces0−→V i−→˜A j−→I R−→0being an exact sequence of Lie algebras.Proof.If the exact sequence is one of Lie algebras,we of course have a Lie algebra structure on V and the map D determined by D a v=[i(a),i(v)]satisﬁes all requirements to deﬁne a Lie algebra structure on A.Conversely,assume we have a Lie algebra structure on the aﬃne space A.If weﬁx an element a∈A,then every element z∈˜A can be written in the form z=λi(a)+i(v).We can deﬁne a bracket of two elements z1=λ1i(a)+i(v1) and z2=λ2i(a)+i(v2)by[z1,z2]=i([v1,v2]+λ1D a v2−λ2D a v1)This bracket is clearly bi-linear and skew-symmetric,and a straightforward cal-culation shows that it satisﬁes the Jacobi identity.Moreover,the deﬁnition does not depend on the choice of the point a;if a′is another point in A,then a′=a+w for some w∈V and the compatibility condition implies that the result is independent of that choice.Finally,it is obvious that the maps i and j then are Lie algebra homomorphisms.Notice that the only condition for a Lie algebra structure on˜A to be an extension of I R by V is that the bracket takes values in V,symbolically:[˜A,˜A]⊂V.Once we have chosen an aﬃne frame on A,we have that the bracket on˜A is determined by the brackets of the associated basis elements.These must be of the form[e0,e0]=0[e0,e j]=C k0j e k[e i,e j]=C k ij e k,since all brackets must take values in the image of the map i.It is well known that a Lie algebra structure on a vector space deﬁnes, and is deﬁned by,a linear Poisson structure on the dual vector space.In the light of the results of the previous section we have a Poisson bracket on V∗and one on A†.Furthermore,˜A being an extension of I R by V,we have that the Poisson structureΛA†is an extension byΛV∗ofΛI R=0(see[2]for the details on Poisson extensions and their relations to Lie algebra extensions). Therefore,once we haveﬁxed a point a∈A,we have a splitting of the sequence 0−→V i−→˜A j−→I R−→0given by h(λ)=λi(a),and the Poisson tensorcan be written asΛA†=ΛV∗+X a∧X Da ,where X Dais the linear vectorﬁeld associated to the linear map D a∈End(V),and X a is the constant vector corresponding to i(a).Moreover,we have that L XD aΛV∗=0.In the coordinatesµ0,µ1,...,µn on A†associated to the basis{eα},we have {µ0,µ0}=0{µ0,µj}=µk C k0j{µi,µj}=µk C k ij,6where C k0j,C k ij are the structure constants introduced above.Therefore,the Poisson tensor readsΛ†A=1∂µi∧∂∂µ0∧∂Then,conditions1and2in the deﬁnition above are trivially satisﬁed.More-over,if weﬁx a0∈A and write a i=a0+v i,then by skew-symmetry of˜ωwe haveω(a1,...,a k)=˜ω(i(a1),...,i(a k))=˜ω(i(a0)+i(v1),...,i(a0)+i(v k))=kj=1(−1)j+1˜ω(i(a0),i(v1),..., i(v j),...,i(v k))+˜ω(i(v1),...,i(v k))=kj=1(−1)j+1ω0(a0,v1,..., v j,...,v k)+ω(v1,...,v k),which proves condition3.Conversely,assume we are given a k-formωon the aﬃne space A with its associatedω0andω.Fixing a0∈A,we know that every point z∈˜A can be written in the form z=λi(a0)+i(v)forλ∈I R and v∈V.We deﬁne the map ˜ωby˜ω(z1,...,z k)=˜ω(λ1i(a0)+i(v1),...,λk i(a0)+i(v k))=kj=1(−1)j+1λjω0(a0,v1,..., v j,...,v k)+ω(v1,...,v k).By virtue of1and3,it follows that˜ωis multi-linear and skew-symmetric, i.e.it is a k-form on˜A.Moreover,˜ω(z1,...,z k)is independent of the choice of the point a0.Indeed,if we choose a diﬀerent point a′0=a0+w,then z j=λj i(a′0)+v′j with v′j=v j−λj w,and applying the deﬁnition above we get ˜ω(z1,...,z k)=˜ω(λ1i(a′0)+i(v′1),...,λk i(a′0)+i(v′k))=kj=1(−1)j+1λjω0(a′0,v′1,..., v′j,...,v′k)+ω(v′1,...,v′k)=kj=1(−1)j+1λjω0(a0+w,v1−λ1w,..., v j,...,v k−λk w) +ω(v1−λ1w,...,v k−λk w)=kj=1(−1)j+1λjω0(a0,v1,..., v j,...,v k)+ω(v1,...,v k)where we have used the properties ofω0andω.The form˜ωis unique,since,if˜θis a k-form on˜A such that i∗˜θ=0,then it follows that the associatedθ0andθvanish from where we deduce that˜θ=0.Once a reference frame has beenﬁxed on A,a1-form˜ωon˜A is of the form ˜ω=ω0e0+ωi e i,and then the local representation ofω=i∗˜ωlooks exactly the same.8More generally,a p-form˜ωon˜A is of the form˜ω=1(p−1)!ω0i1···i p−1e0∧e i1∧···∧e i p−1+1Deﬁnition3.A Lie algebroid structure on E consists of a Lie algebra structure on the(real)aﬃne space of sections of E together with an aﬃne mapρ:E→T M(the anchor),satisfying the following compatibility conditionDσ(fζ)=ρ(σ)(f)ζ+fDσζ,for everyσ∈Sec(E),ζ∈Sec(V)and f∈C∞(M),and where Dσis the action σ→Dσof Sec(E)on Sec(V).The compatibility condition ensures that the associationσ→Dσ,which acts by derivations on the real Lie algebra Sec(V),also acts by derivations on the C∞(M)-module Sec(V).The anchor mapρextends to a linear map˜ρ:˜E→T M,which we will describe in more detail below.It is of interest,however,to observe now already that the map i:V→˜E is a morphism of Lie algebroids,since we have[i(η1),i(η2)]=i([η1,η2])and˜ρ◦i=ρ,whereρis the linear part ofρ.On the contrary,the map j:˜E→M×I R is not a morphism of Lie algebroids,since we have that j([ζ1,ζ2])=˜ρ(ζ1)f2−˜ρ(ζ2)f1, while[j(ζ1),j(ζ2)]=0since theﬁbres of M×I R are1-dimensional.The aﬃne Lie algebroid structure we studied in[20]is the case that M is furtherﬁbred over the real lineπ:M→I R and the anchor mapρtakes values in J1M.Notice that such an extraﬁbration is not generally available,not even locally.For instance,if we take any aﬃne bundleτ:E→M with the trivial Lie algebroid structure(null bracket and anchor)then there is noﬁbration over I R such that the image ofρis in J1M,since the vectors in i(J1M)are non-zero.The following result shows that one can alternatively deﬁne an aﬃne Lie algebroid structure on E as a vector Lie algebroid structure([,],˜ρ)on˜E such that the bracket of two sections in the image of i belongs to the image of i. Theorem2.A Lie algebroid structure on the vector bundle˜τ:˜E→M which is such that the bracket of sections in the image of i lies in the image of i deﬁnes a Lie algebroid structure on the aﬃne bundleτ:E→M,whereby the brackets and maps are determined by the following relations:i([η1,η2])=[i(η1),i(η2)]i(Dση)=[i(σ),i(η)]ρ(σ)=˜ρ(i(σ)).Conversely,a Lie algebroid structure on the aﬃne bundleτ:E→M extends to a Lie algebroid structure on the vector bundle˜τ:˜E→M such that the bracket of sections in the image of i is in the image of i.If weﬁx a sectionσof E and write sectionsζof˜E(locally)in the formζ=fi(σ)+i(η)then the anchor and the bracket are given by˜ρ(ζ)=fρ(σ)+ρ(η)[ζ1,ζ2]= ˜ρ(ζ1)(f2)−˜ρ(ζ2)(f1) i(σ)+i [η1,η2]+f1Dση2−f2Dση1 .10Proof.The veriﬁcation of the above statements is straightforward but rather lengthy.We limit ourselves to checking that the compatibility conditions be-tween brackets and anchors are satisﬁed.For theﬁrst part,weﬁndi Dσ(fη) =[i(σ),i(fη)]=[i(σ),f i(η)]=˜ρ(i(σ))(f)i(η)+f[i(σ),i(η)]=i ρ(σ)(f)η+fDσ(η) , form which it follows that Dσ(fη)=ρ(σ)(f)η+fDσ(η).For the converse,observe that[ζ1,fζ2]−f[ζ1,ζ2]=f2˜ρ(ζ1)(f)i(σ)+i f1ρ(σ)(f)η2+ρ(η1)(f)η2=˜ρ(ζ1)(f)f2i(σ)+(f1ρ(σ)+ρ(η1))(f)i(η2)=˜ρ(ζ1)(f)ζ2,which is the required compatibility condition.In coordinates,if x i denote coordinates on M and yαﬁbre coordinates on E with respect to some local frame(e0,{eα})of sections of E,then we haveρ(e0+yαeα)=(ρi0+ρiαyα)∂∂x i,and the bracket is determined by[e0,e0]=0[e0,eβ]=Cγ0βeγ[eα,eβ]=Cγαβeγ.As aﬁnal remark we mention that the orbits of the Lie algebroidτ:V→M are subsets of the orbits of the algebroid˜τ:˜E→M and they are equal if and only if there exists a sectionσof E such thatρ(σ)is in the image ofρ.6Exterior diﬀerential and Poisson structure Now that we have proved that a Lie algebroid structure on an aﬃne bundle is equivalent to a Lie algebroid structure on˜E,we can deﬁne the exterior diﬀeren-tial operator on E by pulling back the exterior diﬀerential on˜E.More precisely, given a k-formωon the aﬃne bundle E we know that there exists a unique˜ωon˜E such thatω=i∗(˜ω).Then we deﬁne dωas the(k+1)-form given bydω=i∗(d˜ω).11It is easy to see that this deﬁnition is equivalent to the one given in[20](at least when M isﬁbred over I R).The deﬁnition given here has some clear advantages.For instance,the prop-erty d2=0which was rather diﬃcult to prove in[20],becomes evident now:d2ω=d(d(i∗˜ω))=d(i∗d˜ω)=i∗(d2˜ω)=0.The diﬀerential d on the Lie algebroid V is also related to the diﬀerential on˜E;for every k-form˜ωon E we have thatdi∗˜ω=i∗(d˜ω),which in fact simply expresses that i is a morphism of Lie algebroids.Hence, one canﬁnd the diﬀerential of a formωon V by choosing a k-form˜ωon˜E such that i∗˜ω=ωand then obtain dωas i∗(d˜ω).That this does not depend on the choice of˜ωis expressed in equivalent terms by the following result.Proposition4.The ideal I={˜ω|i∗˜ω=0}is a diﬀerential ideal,i.e.d I⊂I.Proof.If i∗˜ω=0,then the same is true for d˜ωsince i∗(d˜ω)=di∗(˜ω)=0.The ideal I is generated by the1-form e0,I={e0∧θ|θis a form on˜E} so that de0belongs to I.The following result shows that e0in fact is d-closed and that this property characterises aﬃne structures.Theorem3.A Lie algebroid structure on˜E restricts to a Lie algebroid struc-ture on the aﬃne bundle E if and only if the exterior diﬀerential satisﬁes de0=0.Proof.Indeed,taking two sectionsσ1andσ2of E we havede0(i(σ1),i(σ2))=˜ρ(σ1) e0,i(σ2) −˜ρ(σ2) e0,i(σ1) − e0,[i(σ1),i(σ2)]=− e0,[i(σ1),i(σ2)]It follows that[i(σ1),i(σ2)]is in Im(i)=Ker(e0)if and only if de0vanishes on the image of i,which spans˜E.In coordinates,the exterior diﬀerential operator is determined byd f=ρi0∂f∂x ieα,for f∈C∞(M)andde0=0,deγ=−Cγ0βe0∧eβ−1f cannot simultaneously vanish,hence f deﬁnes a localﬁbration and then for any sectionσof E we have thatρ(σ)f= d f,σ = e0,σ =1,which is the condition for the anchor being1-jet-valued.When we have a Lie algebroid structure on˜E,there is a Poisson bracket on the dual bundle˜E∗=E†.Any sectionζof˜E,and in particular any section of E,determines a linear function ζon E†byζ(ϕ)= ζm,ϕ for everyϕ∈E†m.Then the Poisson bracket is determined by the condition{ ζ1, ζ2}= [ζ1,ζ2],which for consistency(using linearity and the Leibnitz rule)requires that we put{ ζ,g}=˜ρ(ζ)(g),and{f,g}=0,for f and g functions on M.It is of some interest to mention yet another characterisation of the result described in Theorem2.The above Poisson bracket in fact is determined by the bracket of linear functions coming from sections of E,since these span the set of all linear functions on E†.But the bracket of sections of E is a section of V;it follows that the corresponding Poisson brackets are independent of the coordinateµ0,and therefore,∂∂µ0∧X0,with X0=ρ(e0).(Noticethatρ(e0)=ρ(σ)for any sectionσof E.)The other brackets vanish,so there are no further conditions.In coordinates,we have{x i,x j}=0{µ0,x i}=ρi0{µα,x i}=ρiα{µ0,µβ}=Cγ0βµγ{µα,µβ}=Cγαβµγ13and therefore the Poisson tensor isΛE†=ρiα∂∂x i+1∂µα∧∂∂µ0∧ ρi0∂∂µβ .7ExamplesThe canonical aﬃne Lie algebroid The canonical example of a Lie alge-broid over an aﬃne bundle is theﬁrst jet bundle J1M→M to a manifold M ﬁbered over the real lineπ:M→I R.The elements of the manifold J1M are equivalence classes j1tγof sectionsγof the bundleπ:M→I R,where two sec-tions are equivalent if they haveﬁrst order contact at the point t.It is an aﬃne bundle whose associated vector bundle is Ver(π)the set of vectors tangent to M which are vertical over I R.In this case it is well-known that J1M†=T∗M, and thereforeJ1M=T M.The canonical inmersion is given byi(j1tγ)=˙γ(t),i.e.it maps the1-jet of the sectionγat the point t to the vector tangent toγat the point t.In coordinates,if j1tγhas coordinates(t,x,v)theni(t,x,v)=∂∂x i (t,x)An element w of the associated vector bundle Ver(π)is of the formw=w i∂∂t +X i(t,x)∂∂x iwhich is obviously a section of the vector bundle.Aﬃne distributions An aﬃne E sub-bundle of J1M is involutive if the bracket of sections of the sub-bundle is a section of the associated vector bundle. Therefore,taking as anchor the natural inclusion into T M and as bracket the restriction of the bracket in J1M to E we have an aﬃne Lie algebroid structure on E.14Lie algebra structures on aﬃne spaces We consider the case in which the manifold M reduces to one point M={m}.Thus our aﬃne bundle is E={m}×A and the associated vector bundle is W≡{m}×V for some aﬃne space A over the vector space V.Then,a Lie algebroid structure over the aﬃne bundle E is just an aﬃne Lie algebra structure over A.Indeed,every section of E and of W is determined by a point in A and V,respectively.The anchor must vanishes since T M={0m},so it does not carry any additional information. Trivial aﬃne algebroids By a trivial aﬃne space we mean just a point A={O},and the associated vector space is the trivial one V={0}.The extended aﬃne dual of A is A†=I R since the only aﬃne maps deﬁned on a space of just a point are the constant maps.It follows that the extended bi-dual is˜A=I RGiven a manifold M,we consider the aﬃne bundle E=M×{O}with associated vector bundle V=M×{0}.On V we consider the trivial bracket [,]=0and the anchorρ=0,and as derivation D O we also take D O=0.Now, to construct a Lie algebroid structure on E,we take an arbitrary vectorﬁeld X0on M as given and deﬁne the mapρ:E→T M byρ(m,O)=X0(m).Then it follows thatρis compatible with D O.The extended dual of E is E†=M×I R and the extended bi-dual is˜E= M×I R.We therefore have one section e0spanning the set of sections of E†, and the dual element e0(which is just the image under the canonical immersion of the constant section of value0.)We want to study the associated exterior diﬀerential operator and Poisson bracket.For the exterior diﬀerential operator,since we have1-dimensionalﬁbre on E†it follows that de0=0.On functions f∈C∞(M)we have d f=ρ(e0)(f)e0= X0(f)e0.For the Poisson structure,since theﬁbre of E†is1-dimensional,it is deter-mined by the equation{ˆe0,f}=ρ(e0)(f)andˆe0=µ0.We have that the only non-trivial brackets are{µ0,f}=X0(f).Therefore,the Poisson tensor is∂Λ=sectionsσi(m)=(m,ξi),is the constant section corresponding to the bracket of the values[σ1,σ2](m)=(m,[ξ1,ξ2]A).If we consider the Lie algebra˜A then˜A acts also on the manifold M.The extension˜E is the Lie algebroid associated to the action of˜A.Poisson manifolds with symmetry Consider a Poisson manifold(M,Λ) and an inﬁnitesimal symmetry Y∈X(M)ofΛ,that is L YΛ=0.Take E to be T∗M with its natural aﬃne structure,where the associated vector bundle is V=T∗M itself.On V we consider the Lie algebroid structure deﬁned by the canonical Poisson structure.For a sectionαof E(i.e.a1-form on M)we deﬁne the map Dα:Sec(V)→Sec(V)byDαβ=L Yβ+[α,β].Since Y is a symmetry ofΛ,Dαis a derivation and clearly satisﬁes the required compatibility condition.If we further consider the aﬃne anchorρ:E→T M, determined byρ(αm)=Λ(αm)+Y m,then we have a Lie algebroid structure on the aﬃne bundle E.In this case,since there is a distinguished section of E(the zero section),we have that E†=T M×I R and˜E=T∗M×I R.Jets of sections in a groupoid Let G be a Lie groupoid over a manifold Mwith sourceαand targetβ(the notation is as in[1]).Let TαG(0)G=ker Tα|G(0)be the associated Lie algebroid,that is,the set ofα-vertical vectors at points in G(0)(the set of identities).The anchor is the mapρ=Tβ.Assume that M is furtherﬁbred over the real line,π:M→I R and consider the bundleE=JαG(0)G of1-jets of sections ofπ◦βwhich areα-vertical,at points in G(0).This is an aﬃne bundle whose associated vector bundle is(TαG(0)G)ver the set of(π◦β)-vertical vectors on TαG(0)G.If i is the natural inclusion of(TαG(0)G)ver intoTαG(0)G and we deﬁne the map j:TαG(0)G→M×I R by j(v)=(α(τG(v)),t(v))(where t=π◦β◦τG),then we have the exact sequence of vector bundles over M0−→(TαG(0)G)ver i−→TαG(0)G j−→M×I R−→0and j−1(M×{1})=JαG(0)G.Moreover,the bracket of two sections of JαG(0)G isvertical over I R from where it follows that the Lie algebroid structure of TαG(0)Grestricts to a Lie algebroid structure on the aﬃne bundle JαG(0)G.8ProlongationIn this section we deﬁne the prolongation of aﬁbre bundle with respect to a (vector)Lie algebroid.We are primarily interested in the prolongation of the bundle E→M,but we will describe explicitly a more general constructionﬁrst, since this does not introduce extra complications(see also[8]for generalities).16。

On quasicrystal Lie algebras

On quasicrystal Lie algebrasVolodymyr Mazorchuk2000Mathematics Subject Classiﬁcation:17B68,17B10,17B81Key words:aperiodic Virasoro algebra,highest weight module,Shapovalov form,Kac determinantAbstractWe realize the aperiodic Witt and Virasoro algebras as well as other quasicrystal Lie algebras as factoralgebras of some subalgebras of the higher rank Virasoro alge-bras.This realization allows us to generalize the notion of quasicrystal Lie algebras.In the case when the constructed algebra admits a conjugation,we compute the Kacdeterminant for the Shapovalov form on the corresponding Verma modules.In thecase of the aperiodic Virasoro algebra this proves the conjecture of R.Twarock.1IntroductionThis paper has grown up from my attempt to understand the recently introduced notion of quasicrystal Lie algebras and the aperiodic Witt and Virasoro algebras,[PPT,T1]. These algebras form a new family of inﬁnite-dimensional Lie algebras,whose generators are indexed by points of an aperiodic set(which is in fact a one-dimensional cut-and-project quasicrystal,an object,intensively studied by many authors,see e.g.[Ka,K,R] and references therein).Quasicrystal Lie algebras and their representations were studied in[PT,PPT,T1]and in[T2,T3]some applications of these algebras to construction of some integrable models in quantum mechanics were given.However,there are many important questions about the quasicrystal Lie algebras,which are still open.For example,in[T1]the author constructs a triangular decomposition for the aperiodic Virasoro algebra,hence constructing Verma modules,and conjectures a formula for the Kac determinant of the Shapovalov form on these modules.This formula in important both for description of simple highest weight modules and for picking up those of them which can be unitarizable,which is the question of primary interest in physical applications.It was clear from the veryﬁrst deﬁnition of quasicrystal Lie algebras,that this notion should be closely connected with the notion of the higher rank Virasoro algebras,deﬁned in [PZ].The major diﬀerence between these algebras is that the indexing set for quasicrystal Lie algebras is a discreet subset of R while for the higher rank Virasoro algebras the corresponding set is everywhere dense.In the present paper we establish this connection1by realizing quasicrystal Lie algebras as factoralgebras of some subalgebras of the higher rank Virasoro algebras.This realization allows us to generalize quasicrystal algebras in several directions,preserving the property to have a discreet indexing set.Moreover,the notion and construction of triangular decomposition for these algebras appears naturally in this framework.Further,we discuss the existence of conjugation on constructed algebras, which pairs the components of the positive and negative part.In the case,when such pairing exists,the deﬁnition of the Shapovalov form on Verma modules(see[S,MP])is straightforward and we compute the Kac determinant(see[S,KK,MP,KR])of this form. In the case of the aperiodic Virasoro algebras this proves[T1,Conjecture V.7].The paper is organized as follows:in Section2and Section3we remind the deﬁnitions of quasicrystal Lie algebras and higher rank Virasoro algebras.We give a realization of quasicrystal Lie algebras as factoralgebras of certain subalgebras of the higher rank Virasoro algebras in Section4and use it to construct parabolic and triangular decompositions of our algebras in Section5.In Section6we study the Verma modules and,in particular, calculate the determinant of the Shapovalov form on them.Weﬁnish with discussing several generalizations of our construction in Section7and Section8.2Quasicrystal Lie algebras√Denote by(·)#the unique non-trivial automorphism of theﬁeld Q(5.LetΩbe a non-empty,connected and bounded real set, whose set of inner points does not contain0.Setτ=15).Then the quasicrystal Σ(Ω),associated withΩ,is the set of all x∈Z[τ],such that x#∈Ω.The quasicrystal Lie algebra L(Ω),associated withΩ,is deﬁned as follows(see[PPT]):it is generated over F by L x,x∈Σ(Ω),with the Lie bracket deﬁned via[L x,L y]= (y−x)L x+y,x+y∈Σ(Ω)0,otherwise.To deﬁne the aperiodic Witt and Virasoro algebras as it is done in[T1],we introduce the mapϕ:Z[τ]→Z,which sends x=a+bτtoϕ(x)=b.Then the aperiodic Witt algebra AW([0,1],F)is generated over F by L x,x∈Σ([0,1]),with the Lie bracket deﬁned via[L x,L y]= (ϕ(y)−ϕ(x))L x+y,x+y∈Σ([0,1])0,otherwise.By[T1,Theorem III.4],the algebra AW([0,1],F)admits the unique central extension AV([0,1],F),called the aperiodic Virasoro algebra,which is generated over F by L x,x∈Σ([0,1]),and c,with the Lie bracket deﬁned via[L x,L y]= (ϕ(y)−ϕ(x))L x+y+δϕ(x),−ϕ(y)ϕ(x)3−ϕ(x)33The higher rank Virasoro algebrasLet P denote the free abelian group Z k ofﬁnite rank k andψ:P→(F,+)be a group monomorphism.The rank k Virasoro algebra V(ψ,F),associated withψ,is generated over F by elements e x,x∈P,and central c,with the Lie bracket deﬁned viaψ(x)3−ψ(x)3[e x,e y]=(ψ(y)−ψ(x))e x+y+δx,−yadmissible order on P.Indeed,<is obviously antisymmetric,antireﬂexiv and transitive.So,it is a partial order.But from the deﬁnition it also follows immediately,that<is linear. Further,for any a<b in P and c∈P we have(ψ(a+c)−ψ(b+c))#=(ψ(a+c−b−c))#=(ψ(a−b))#=(ψ(a)−ψ(b))#<0and hence a+c<b+c,thus<is compatible withthe addition in P.Finally,if0<a<b then0<(ψ(a))#and hence there always exists k∈N such that(ψ(b)−ψ(ka))#=(ψ(b))#−k(ψ(a))#<0,which shows that the order is admissible.Consider the rank2Witt algebra G=W(ψ,F).Without loss of generality we can assume thatΩ⊂R0,as in other case we can work with the order,opposite to<.AsΩis a connected bounded subset of R,it has one of the following four forms:[a,b],(a,b],[a,b), (a,b)for some non-negative real a,b.We deﬁne I and J as follows:I is generated by all e x such thatψ(x)#>a(resp.ψ(x)# a)if a∈Ω(resp.a∈Ω);and J is generated by all e x such thatψ(x)#>b(resp.ψ(x)# b)if b∈Ω(resp.b∈Ω).From the deﬁnition it follows immediately that both I and J are non-negative ideals of P with respect to <.Hence,the algebras L(G,<,I)and L(G,<,J)are well-deﬁned subalgebras of G and L(G,<,J)⊂L(G,<,I)by deﬁnition.Now we show that L(G,<,J)is actually a ideal of L(G,<,I).Indeed,if x∈I andy∈J we get that x+y∈J as J is an ideal of P and x∈P0+.Hence[e x,e y]∈L(G,<,J) for any e x∈L(G,<,I)and e y∈L(G,<,J).Finally,we consider the map f:L(G,<,I)→L(Ω)deﬁned byf(e x)= L x,x ∈Ω0,otherwise.From the deﬁnition of the Lie brackets in L(Ω)(Section2)and in G(Section3)weimmediately get that f is a Lie algebra homomorphisms.Moreover,it is also clear that its kernel coincides with L(G,<,J).This completes the proof.Theorem1motivates the following deﬁnition:let G=W(P,ψ)be a higher rank Witt algebra(it is important here that k>1,i.e.that G is not the classical Witt algebra), <be an admissible order on P,and I⊃J be two non-negative ideals of P with respect to the order<.Then we deﬁne the Lie algebra A(P,ψ,<,I,J)of quasicrystal type as the quotient algebra L(G,<,I)/L(G,<,J).In particular,all quasicrystal Lie algebras are Lie algebra of quasicrystal type.Now we can formulate some basic properties of Lie algebras of quasicrystal type and we see that these algebras share a lot of properties of classical quasicrystal Lie algebras.We start with the following easy observation.Lemma 1.Let<be an admissible order on Z k.Then there exists a homomorphism,σ:P→R,such thatσ(P±)⊂R±.Proof.Identify P with Z k⊂R ing the description of admissible orders on an abelian group from[Z],weﬁnd a hyperplane,H,of R k,such that P+coincides with the set of points from Z k,which are settled on the same side with respect to H.Thenσcan be taken,e.g.the projection on H⊥with respect to H(here R k is considered as an Euclidean space in a natural way).4For given G=W(P,ψ)and<weﬁx someσ,existing by Lemma1.We deﬁne a= inf x∈I(σ(x))and b=inf x∈J(σ(x))and will use this notation in the following statement. Proposition1.Let A(P,ψ,<,I,J)be a Lie algebra of quasicrystal type.1.A(P,ψ,<,I,J)is abelian if and only if2a b.2.A(P,ψ,<,I,J)has non-trivial center if and only if a=0.3.A(P,ψ,<,I,J)is nilpotent if and only if a>0and J=∅.4.The algebra A(P,ψ,<,I,J)is perfect if an only if a=0and0∈I.5.If J=∅then anyﬁnite set of elements in A(P,ψ,<,I,J)generates aﬁnite-dimensional Lie subalgebra of A(P,ψ,<,I,J).In particular,A(P,ψ,<,I,J)has ﬁnite-dimensional subalgebras of arbitrary non-negative dimension.Proof.All statements are easy corollaries from the additivity of indices of generating ele-ments under the Lie bracket.Indeed,with this remark theﬁrst statement reduces to the fact that x a and y a implies x+y 2a b;the second one reduces to the fact that for any x a and y>b−a holds x+y>b;and the third one reduces to the fact that for ka>b we have kx>b for any x a.If a=0,the algebra A(P,ψ,<,I,J)is nilpotent by statement three and hence not perfect.It is also clear that it is impossible to get0as a result of the Lie operation.But if a=0and0∈I,then L(G,<,I)=G+,σ(P+)is dense in R+and hence for any x>0there are y,z∈P+such that y+z=x.This implies that A(P,ψ,<,I,J)is perfect in this case and hence the property four.Theﬁrst part of the last statement is equivalent to the trivial statement that aﬁnite subset of R+generates an additive semigroup,whose intersection with any bounded set is ﬁnite.To prove the second part it is suﬃcient to consider the span of e ix,i=1,...,n, such that nx<b and(n+1)x>b.This completes the proof.For example,to realize the aperiodic Witt algebra AW([0,1]),deﬁned in[T1],as a Lie algebra of quasicrystal type,one should take P=Z2,ψbeing the projection on the second coordinate;<deﬁned by x<y if and only if the inner product of y−x with(1,15)) is greater than zero;I=P0+;J={x∈P:(1,0)<x}.5Standard and non-standard triangular decomposi-tionsThe realization of Lie algebras of quasicrystal type,obtained in the previous section al-lows us to adopt the technique from[M]to construct various triangular and parabolic decompositions of these algebras.The general procedure will look as follows.Let A=A(P,ψ,<,I,J)be a Lie algebra of quasicrystal type.Abusing notation we will denote by e x,x∈I\J,the generators of A.Choose any linear pre-order, ,on the abelian5group P ,which is diﬀerent from <and its opposite.Deﬁne A ±as the Lie subalgebras of A ,generated by all e x ,0≺±x ,and set A 0to be the Lie subalgebra of A ,generated by all e x ,0 x and x 0.We get the following obvious fact.Lemma 2.A =A −⊕A 0⊕A +.Proof.Clearly,A =A −+A 0+A +.The fact that this is actually a direct sum decomposition follows easily from the the property x y implies x +z y +z .It is natural to call the decomposition A =A −⊕A 0⊕A +parabolic decomposition of A ,associated with .Given a parabolic decomposition and a simple A 0-module,V ,one can extend V to an A 0⊕A +-module with the trivial action of A +and construct the associated generalized Verma module M (V )as follows:M (V )=U (A )⊗U (0⊕+)V .If A 0happens to be rather special,it is natural to rename the corresponding parabolic decomposition into triangular decomposition .However this is a subtle question and the hierarchy I give here represent only my point of view and is inspired by the corresponding notions for the higher rank Virasoro algebra ([M]).We will say that the decomposition A =A −⊕A 0⊕A +is a standard triangular de-composition provided A 0=F e x for some x ∈P ,which is not maximal in I \J .We call A =A −⊕A 0⊕A +the non-standard triangular decomposition provided A 0is a commutative Lie algebra and the parabolic decomposition fails to be a standard triangular.In the case of triangular decomposition generalized Verma modules become classical Verma modules as in this case dim(V )=1.The ﬁrst case is natural and corresponds to triangular decompositions of the higher rank Virasoro algebras,[M].Actually,here one has to be careful because,depending on whether ≺satisﬁes the Archimed law or not,one can further distinguish two cases of standard triangular decomposition.We will not do this,as we will not study the diﬀerence between the corresponding situations.But the second case has a striking diﬀerence from the ﬁrst one and comes from the deﬁnition of triangular decomposition for the aperiodic Virasoro algebra in [T1].This means that the triangular decomposition for the aperiodic Virasoro algebra,constructed in [T1]is an example of a non-standard triangular decomposition.We now will study analogous situations in more detail.We retain the notation for σ,a,b from the previous section and further assume that a =0and that that there is an element,e u ∈A ,such that σ(u )=b and ψ(u )=0.Since <is an admissible order,such element is unique and we retain the notation e u for it.Lemma 3.Under the above assumptions we consider the vectorspace A =A ⊕F c .Then the formula [e x +a c ,e y +b c ]=[e x ,e y ]+ψ(x )3−ψ(x )The algebra A,constructed in Lemma3is a natural generalization of the aperiodic Virasoro algebra from[T1].In particular,the aperiodic Virasoro algebra coincides with A for A constructed in the end of the previous section.However,if e.g.the rank of P is bigger than two,we get an example of A ,which diﬀers from the aperiodic Virasoro algebra.We will call algebras A the Virasoro-like algebras of quasicrystal type.Assigning the element c index u we easily transfer the notions of parabolic and both standard and non-standard triangular decompositions on algebra A .If P has rank two, then,up to taking the opposite order,the non-standard triangular decomposition of A, such that A 0contains e u,is unique,and in the case of the aperiodic Virasoro algebra this coincides with the triangular decomposition,constructed in[T1,Section V].For the rank two case once can easily construct example of A such that with respect to the unique natural non-standard triangular decomposition,mentioned above,dim(A 0)is an arbitrary positive integer.Hence even in rank two case one gets a lot of examples of A ,diﬀerent from the aperiodic Virasoro algebras.All these algebras will have discrete aperiodic root systems,and,if considered as graded by the action of e0,all roots will be multiple with multiplicity dim(A 0)−2.In the case of the aperiodic Virasoro algebra we have dim(A 0)=3 and hence all roots(with non-zero action of e0)are multiplicity free.We will discuss this situation in more details in the next section,when we will deﬁne the Shapovalov Form on the Verma modules and compute its determinant.6Shapovalov form and Kac determinantIn this section we present several results on the structure of Verma modules over Lie and Virasoro-like algebras of quasicrystal type.As in the case of the Witt and the Virasoro algebras,the representation theory the last one is more complicated,which,in particular, gives a bigger variety of simple highest weight modules.Our main tool in the case of the Virasoro-like algebras of quasicrystal type and the corresponding Lie algebras of quasicrys-tal type will be the Shapovalov form on Verma modules,ﬁrst deﬁned in[S]for simple ﬁnite-dimensional Lie algebras.However,we start with more elementary general case of Lie algebras of quasicrystal type,which happens to be really trivial.Before starting we just note that in this section we always assume that F is an algebraically closedﬁeld of characteristic zero.We recall that,given a triangular decomposition,A=A−⊕A0⊕A+,an A-module,M, is called a highest weight module,if there exists a generator,v∈M,such that A+v=0. Proposition2.All simple highest weight modules over a Lie algebras of quasicrystal type, which correspond to a standard triangular decompositions with A0=F e x⊂[A,A]are one-dimensional.In particular,corresponding Verma modules are always reducible. Proof.This is a direct corollary of A0=F e x and e x∈[A,A].Proposition3.All Verma modules over a Lie algebras of quasicrystal type,which corre-spond to a standard triangular decompositions with A0=F e x⊂[A,A],are reducible.The7corresponding unique simple quotients are one-dimensional if and only if the eigenvalue ofe x on the primitive generator of the module is zero.Otherwise they are inﬁnite-dimensional. Proof.Let v be the canonical generator of the Verma module in question.The reducibilityfollows from the fact that x is not maximal in I\J,and hence there are inﬁnitely manyelements y∈P−satisfying e x∈[e y,A],which implies that U(A)e y v is a proper submodule of the Verma module.The second statement follows considering the set of elements e y,y∈P−,satisfying e x∈[e y,A],which is obviously inﬁnite.So,we can now move on to the case of non-standard triangular decomposition.Firstwe reduce our consideration to the natural case of weight modules withﬁnite-dimensionalweight spaces,which corresponds to the situation,when the root system of A is discrete. This is only possible in the case when P Z2.Here our main tool will be the Shapovalov form and to be able to work with it we will also need the following assumptions from the previous section:e0∈A;and there is e u∈A,such thatσ(u)=b andψ(u)=0.As it was mentioned above,this situation covers,for example,the case of the aperiodic Witt algebra.Since in the case of the algebra A the arguments will be absolutely the same,we consider both cases simultaneously with all the notation for the algebra A .The case of A is then easily obtained by factoring c=0out.We deﬁne the conjugation on P viaω(x)=u−x and it follows immediately from our assumptions that e x∈A implies eω(x)∈A .However,is easily to see thatωdoes not extend to an(anti)involution on A .We note thatσ(ω(x))=b−σ(x).We recall that the algebra A is graded by the adjoint action of e0(or,more general,A 0)and for C α=0the dimension of A αis either0or dim(A 0)−2(dim(A0)−1inthe case of algebra A).We denote by∆the set of all(non-zero)roots of A with respectto this action and by∆±the sets of all positive and negative roots corresponding to our triangular decomposition.Obviously,ωextends to a linear bijection A α→A −αfor any α∈∆∪{0}.As A 0is commutative,simple A 0-modules are one-dimensional and have the form Vλ,λ∈(A 0)∗,where the action is deﬁned via g(v)=λ(g)v for v∈Vλand g∈A 0.Let vλdenote a canonical generator of M(Vλ).Let∆ (resp.∆ ±)denote the semigroup,generated by∆(resp.∆±).Then the module M(Vλ)is a weight module with respect to A 0with the supportλ∪λ−∆ +.All weight spaces of M(Vλ)areﬁnite dimensional.Moreover,M(Vλ)is isomorphic to U(A −)vλas a vectorspace.The∆±-gradation of A ±extends to the∆ ±-gradation of U(A ±)and,in the aniinvolutive way,ωextends to a linear componentwise isomorphism from U(A +)to U(A −)and back, which matches U(A +)αwith U(A −)−α.Forµ∈Supp(M(Vλ)),µ=λ−ν,ν∈∆ +,we deﬁne the Shapovalov form Fλ,νon M(Vλ)µby setting that Fλ,ν(fvλ,gvλ),f,g∈U(A −)−ν,equals the coeﬃcient ofω(f)gvλ∈M(Vλ)λ,written in the basis{vλ}.The following properties of Fλ=⊕ν∈∆Fλ,νare standard and the reader can consult[KK,MP]for the arguments.8Lemma4. 1.M(Vλ)is simple if and only if Fλis non-degenerate.2.The kernel of Fλcoincides with the unique maximal submodule of M(Vλ).Hence in order to study the reducibility of M(Vλ)it is suﬃcient to compute the determinant of Fλ,νfor allλandν.To be able to do this we consider the followingmonomial generators of U(A −)−ν:G=G(ν)={g(x1,...,x k)=e x1...e xk:x i∈∆−;i x i=−ν;σ(x i) σ(x i+1)}.We deﬁne the linear order on this set of generators as the lexicographical order with respect to the values ofσ(x i).The key property of this construction is the following.Lemma 5.If g(x1,...,x k)∈G and g(y1,...,y m)∈G are such that g(x1,...,x k) g(y1,...,y m).Then Fλ,ν(g(x1,...,x k)vλ,g(y1,...,y m)vλ)=0.Proof.Let i be minimal such thatσ(x i)<σ(y i).Thenσ(ω(x i))>b−σ(y i)and henceeω(xi)commutes with e yiand thus with all e yj,j i,since for such j we haveσ(y j) σ(y i)form the deﬁnition of G.For j<i we have x j=u j and thus[eω(xj),e yj]∈A 0.We canwriteeω(xi)(ω(e x1...e xi−1)e y1...e yi−1)e yi...e ymvλ==ε(ω(e x1...e xi−1)e y1...e yi−1)eω(xi)e yi...e ymvλ+other termsfor someε∈F,where in other terms some eω(xj),j<i,occurs already after the corre-sponding e yj .As eω(xi)commutes with all e yj,j i,we get that theﬁrst summand equalszero.Now consider one of the other terms and let eω(xj)be the factor occurring most to theright in the monomial.This means,in particular,that for s<j this monomial contains[eω(xs),e ys],which are the elements of A 0and thus,up to a scalar factor,can be movedto the left.In particular,σ(ω(x s))is the biggest value among all others occurring in thismonomial.If the element e y,standing next to eω(xj)satisﬁes y=x j,this means that eω(xj)commutes with e y and hence the monomial contributes0to the global sum.Otherwise thenumber of factors,standing to the right from eω(xj),which equal x j,is less than the samenumber before the last commutation.Hence induction in this number reduces the problem to the case y=x j thus proving that all monomials occurring in other terms contribute0 to the global sum.From this it follows directly that Fλ,ν(g(x1,...,x k)vλ,g(y1,...,y m)vλ)=0,which com-pletes the proof.From Lemma5we immediately get the following statement,which,in particular,proves [T1,Conjecture V.7].Corollary 1.The determinant of Fλ,νcoincides with the product of diagonal elements Fλ,ν(g(x1,...,x k)vλ,g(x1,...,x k)vλ).9Now we can formulate the computation results for the determinant of the Shapovalov form and the corresponding corollaries for the structure of M(Vλ).Denote by P the set of all non-zero x∈P such that e x∈A is non-zero.Then the decomposition∆=∆−∪∆+ induces a decomposition P =P −∪P +.Forν∈∆ +and x∈P −we denote by pν(x)thenumber of occurrences of e x as factors in the canonical decomposition of all monomials in G(ν).Theorem2.Up to a non-zero constant the determinant of Fλ,νequalsx∈P − λ(e u)−ψ(x)2−112c,asψ(u)=0.Moreover,[e u−x,e u]is in fact central in A .Hence,we can movethe non-zero factor2ψ(x)out and get that,up to a non-zero constant factor,we haveFλ,µ(g(x1,...,x k)vλ,g(x1,...,x k)vλ)=ki=1 λ(e u)−ψ(x i)2−1Proof.Under these conditions all factors of the diagonal elements of the matrix of the Shapovalov form are non-negative and hence all leading minors are non-negative as well. This implies the statement.Using these results we also get some information about highest weight modules,asso-ciated with standard triangular decompositions.Corollary4.The dimensions of the weight spaces of inﬁnite-dimensional highest weight modules over Lie algebras of quasicrystal type,associated with standard triangular decom-positions,are not uniformly bounded.Proof.Let A0=F e x be the zero component of the given standard triangular decomposition. Then we can factor our an ideal of A such that the factoralgebra is still of quasicrystal type,but the element x became maximal in the corresponding I\J,and hence the induced triangular decomposition became non-standard.Now we haveλ(e x)=0and hence the corresponding Verma module over this algebra is simple and the dimensions of its weight spaces are obviously unbounded.Buy this module naturally embeds(as a vector subspace) into the simple highest weight module,which we started with.7Further generalizations of quasicrystal Lie algebras Geometrical realization of the algebra A,obtained in Section4,motivates the following generalization of the class of Lie algebras of quasicrystal type.We consider arbitrary rank n Witt algebra G=G(P,ψ)with P Z n being realized in R n in a natural way.LetΩbe a convex subset of R n,containing at least one non-zero point of P,and satisfying the following0-star condition:v∈Ωimpliesλv∈Ωfor all λ>1.In this case we will callΩa0-star sets.Denote by L(Ω)the vectorsubspace in G, spanned by e x,x∈Ω.Lemma6.L(Ω)is a Lie subalgebra of G.Proof.If x,y∈P∩Ωthen x+y=2(12y).12y belongs toΩbecause of theconvexity and thus x+y∈Ωby the0-star condition.Lemma7.LetΩbe a0-star set,v∈Ωandλ>0.Then,if the setΩλ,v=λv+Ωcontains at least one non-zero point of P,it is a0-star set.Moreover,Ωλ,v⊂Ω.Proof.Clearly,Ωλ,v is convex.Further,if w∈Ωandγ>1,thenγ(w+λv)=γ(λ+1)(1λ+1v)belongs toΩby the same arguments as in Lemma6.This completes theproof.Lemma8.LetΩbe a0-star set,v∈Ωandλ>0.Then L(Ωλ,v)is an ideal of L(Ω). Proof.If w∈Ωand w =w +v∈Ωλ,v for some w ∈Ω,then w+w =w+w +v∈Ωλ,v. This implies the statement.11Hence,for arbitrary G,Ωand v as above we can form the algebra A(P,ψ,Ω,λ,v)= L(Ω)/L(Ωλ,v),which we will call a Lie algebra of convex quasicrystal type.To obtain the usual Lie algebra of quasicrystal type,one should takeΩto be a half-space(open or closed), which does not contain0as an inner point.The basic properties of Lie algebras of convex quasicrystal type are similar to those of Lie algebra of quasicrystal type,however,their formulation is much more complicated because it usually depends on the structure ofΩ. Here we list only some most straightforward ones.Proposition4.Let A=A(P,ψ,Ω,λ,v)be a Lie algebra of convex quasicrystal type and a denote the inﬁnum of distances from points inΩ∩P to0.Assume that dim(A)>1and that for any x∈Ωsome neighborhood(in R n)of2x belongs toΩ.Then1.if a>0then any element of A is nilpotent.2.Anyﬁnite set of elements from A generates aﬁnite-dimensional Lie subalgebra of A. Proof.If x∈S=Ω\Ωλ,v then there alway exists y,such that|x−y| |v|and such that y∈Ω.Let w∈Ω.If some ball of radius r over2w belongs toΩ,then,forλ>1the point 2λw belongs toΩtogether with the ball of radiusλr around it.Makingλr>|v|we get that2λw∈Ωλ,v.This implies theﬁrst statement.If the set{w1,...,w k}⊂Ωisﬁnite,then weﬁnd some r such that2w i belongs toΩtogether with its neighbor ball of radius r.Then the same is true for all linear combinations of these elements with non-negative integer coeﬃcients.By the same arguments as in the previous paragraph,there is N∈N such that any linear combination of{Nw1,...,Nw k} with non-negative integer coeﬃcients belongs toΩλ,v.This implies the second statement.Let us study an example of such algebra,which,as we will show,has some interesting properties.Take P=Z2,ψthe projection on the second component,Ω={w∈R2: (w,(1,1)) 0and(w,(1,−1)) 0},v=(n+ ,0),n∈N, ∈(0,1).The corresponding algebra A=A(P,ψ,Ω,λ,v)is graded with respect to the e0action with graded components corresponding to all integers and having dimension n.In particular,one can deﬁne and study triangular(parabolic)decompositions of this algebra and corresponding(generalized) Verma modules.The set P =P∩(Ω\Ωλ,v)coincides with{(a,b):0 a−|b| n}. Deﬁne the conjugationωon this set viaω(a,b)=(2|b|+n−a,−b).Then we have the natural notions of Verma modules and the Shapovalov form on them.In our situation we have∆+=N.Lemma9.The Verma module M(Vλ)is always reducible.However,the unique sim-ple quotient of M(Vλ)is inﬁnite dimensional if and only if at least one of the numbers λ(e(2,0)),...,λ(e(n,0))is non-zero.Otherwise it is one-dimensional.Proof.We note that the intersection of[A,A]with A0coincides with the linear span˜A0 of elements{e(2,0),...,e(n,0)}.Hence,if the restriction ofλon˜A0is zero,the Shapovalov form in identically zero on all M(Vλ)λ−k,k∈N.Otherwise,assume thatλ(e(i,o))=0and take e x∈A1and e y∈A−1such that[e x,e y]= e(i,0).We get Fλ,k(e k,e k x)=0and the statement is proved.ω(y)12。

n-李代数的Hypo-幂零理想

n-李代数的Hypo-幂零理想白瑞蒲;吴万青;时翠梅【摘要】The Hypo-nilpotent ideals of n-Lie algebras and the algebraic structure of solvable 3-Lie algebras which have a 5 dimensional Hypo-nilpotent ideal N are mainly concerned. It is proven that there exist Hypo-nilpotent ideals for the solvable non-nilpotent n-Lie algebras, and the co-dimension of the nilpotent radical is one. The dimensional relations between the maximal Hypo-nilpotent ideals of solvable non-nilpotent 3-Lie algebras and 3-Lie algebras are offered. For a class of 3-Lie algebras which have special a 5 dimensional nilpotent ideal, the dimension sequence DS of derivative algebras and the decreased center the dimension sequence CS and their relations are considered.%研究n-李代数的Hypo-幂零理想,及具有5维极大Hypo-幂零理想的所有可解3-李代数的结构.证明可解非幂零n-李代数一定存在Hypo-幂零理想,且其幂零根基的余维数等于1.给出可解非幂零3-李代数的极大Hypo-幂零理想与3-李代数的维数关系.对具有一类特殊5-维极大次幂零理想的可解3-李代数的每一类3-李代数,分别研究了其导代数维数序列DS及下降中心维数序列CS,及他们之间的关系.【期刊名称】《黑龙江大学自然科学学报》【年(卷),期】2011(028)001【总页数】4页(P4-7)【关键词】n-李代数;Hypo-幂零理想;可解3-李代数【作者】白瑞蒲;吴万青;时翠梅【作者单位】河北大学,数学与计算机学院,保定,071002;河北大学,数学与计算机学院,保定,071002;河北工程大学,理学院,邯郸,056000【正文语种】中文【中图分类】O152.5n－李代数［1－2］(又称为Filippov代数，Nambu－Poisson代数)由前苏联数学家Filippov在1985年提出，但此多元代数系统已经被更早地应用到物理研究领域。

Central extensions of groups of sections

a r X i v :0711.3437v1[mat h.D G]21N ov27Central extensions of groups of sections K.-H.Neeb and Ch.Wockel February 2,2008Abstract If q :P →M is a principal K -bundle over the compact manifold M ,then any invariant symmetric V -valued bilinear form on the Lie algebra k of K deﬁnes a Lie algebra extension of the gauge algebra by a space of bundle-valued 1-forms modulo exact forms.In the present paper we analyze the integrability of this extension to a Lie group ex-tension for non-connected,possibly inﬁnite-dimensional Lie groups K .If K has ﬁnitely many connected components we give a complete char-acterization of the integrable extensions.Our results on gauge groups are obtained by specialization of more general results on extensions of Lie groups of smooth sections of Lie group bundles.In this more gen-eral context we provide suﬃcient conditions for integrability in terms of data related only to the group K .Keywords:gauge group;gauge algebra;central extension;Lie group extension;integrable Lie algebra;Lie group bundle;Lie algebra bundle IntroductionAﬃne Kac–Moody groups and their Lie algebras play an interesting role in various ﬁelds of mathematics and mathematical physics,such as string theory and conformal ﬁeld theory (cf.[PS86]for the analytic theory of loop groups,[Ka90]for the algebraic theory of Kac–Moody Lie algebras and [Sch97]for connections to conformal ﬁeld theory).For further connections to mathe-matical physics we refer to the monograph [Mi89]which discusses various occurrences of Lie algebras of smooth maps in physical theories (see also[Mu88],[DDS95]).1From a geometric perspective,aﬃne Kac–Moody Lie groups can be ob-tained from gauge groups Gau(P)of principal bundles P over the circle S1 whoseﬁber group is a simple compact Lie group K by constructing a central extension and forming a semidirect product with a circle group corresponding to rigid rotations of the circle.Here the untwisted case corresponds to trivial bundles,where Gau(P)∼=C∞(S1,K)is a loop group,and the twisted case corresponds to bundles which can be trivialized by a2-or3-fold covering of S1.In the present paper we address central extensions of gauge groups Gau(P) of more general bundles over a compact smooth manifold M,where the struc-ture group K may be an inﬁnite-dimensional locally exponential Lie group. In particular,Banach–Lie groups and groups of smooth maps on compact manifolds are permitted.Since the gauge group Gau(P)is isomorphic to the group of smooth sections of the associated group bundle,deﬁned by the conjugation action of K on itself,it is natural to address central extensions of gauge groups and their Lie algebras in the more general context of groups of sections of bundles of Lie groups,resp.,Lie algebras.In the following,K always a denotes a locally trivial Lie group bundle whose typicalﬁber K is a locally exponential Lie group with Lie algebra k= L(K).Since we work with inﬁnite-dimensional Lie algebras,we have to face the diﬃculty that,in general,the group Aut(k)does not carry a natural Lie group structure.1Therefore it is natural to consider only Lie algebra bundles which are associated to some H-principal bundle P with respect to a smooth actionρk:H→Aut(k)of a Lie group H on k,i.e.,for which the map(h,x)→ρk(h)(x)is smooth.Let K be such a Lie algebra bundle.Then the smooth compact open topology turns the spaceΓK of its smooth sections into a locally convex topological Lie algebra.To construct2-cocycles on this algebra,we start with a continuous invariant symmetric bilinear mapκ:k×k→Vwith values in a locally convex H-module V on which the identity component H0acts trivially.The corresponding vector bundle V associated to P is ﬂat,so that we have a natural exterior derivative d on V-valued diﬀerential forms.If V isﬁnite-dimensional or H acts on V as aﬁnite group,thend(ΓV)is a closed subspace ofΩ1(M,V),so that the quotientΩ1(M,V),given byω∇κ(f,g):=[κ(f,d∇g)].(1) Here d∇is the covariant exterior diﬀerential onΩ•(M,K)induced by a prin-cipal connection∇on P.For the special case of gauge algebras of principal bundles with connected compact structure group K,cocycles of this form have also been discussed brieﬂy in[LMNS98].Clearly,(1)generalizes the well-known cocycles for Lie algebras of smooth maps,obtained from invari-ant bilinear forms and leading to universal central extensions ofΓK if K is trivial and k is semisimple(cf.[NeWa07],[KL82]).Since we are presently far from a complete understanding of the variety of all central extensions of ΓK or corresponding groups,it seems natural to study this class of cocycles ﬁrst.For other classes of cocycles,which are easier to handle,and their integrability we refer to[Ne08,Sect.4].It seems quite likely that if k isﬁnite-dimensional semisimple,K=Ad(P) is the gauge bundle of a K-principal bundle andκis universal,then the central extension of gau(P)∼=ΓK byΩ1(M,V)3obtained by integration of the left invariant2-form onΓK deﬁned byωis discrete and the adjoint action ofΓK on the central extension ΓK integrates to an action of the corresponding connected Lie group(ΓK)0(cf.Appendix C for more details on these two conditions).Therefore our main task consists in verifying these two conditions,resp.,inﬁnding veriﬁable necessary and suﬃcient conditions for these conditions to be satisﬁed.To obtain information on the period groupΠω,it is natural to compose the cocycle with pullback mapsγ∗:Ω1(S1,γ∗V),deﬁned by smooth loopsγ:S1→M.To make this strategy work,we need quite detailed information on the special case M=S1,for whichΓK is a twisted loop group deﬁned by some automorphismϕ∈Aut(K):ΓK∼=C∞(R,K)ϕ:={f∈C∞(R,K):(∀t∈R)f(t+1)=ϕ−1(f(t))}.The structure of the paper is as follows.In Section1we introduce the cocyclesω∇κand discuss the dependence of their cohomology class on the con-nection∇.In particular,we show that they lift toΩ1(M,V)-valued cocycles ifκis exact in the sense that the3-cocycle C(κ)(x,y,z):=κ([x,y],z)on k is a coboundary(cf.[Ne08],[NeWa07]).In the end of this section we introduce an interesting class of bundles which are quite diﬀerent from adjoint bundles and illustrate many of the phenomena and diﬃculties we encounter in this paper.In Section2we analyze the situation for the special case M=S1,where ΓK is a twisted loop group.It is a key observation that in this case the pe-riod map perωis closely related to the period map of the closed biinvariant 3-form on K,determined by the Lie algebra3-cocycle C(κ).To establish this relation,we need the connecting maps in the long exact homotopy se-quence of theﬁbration deﬁned by the evaluation map ev K0:C∞(R,K)ϕ→K, f→f(0).Luckily,this map is given explicitly in terms of the twistϕand thus can be determined in concrete examples.Having established the re-lation between perωand per C(κ),we use detailed knowledge on per C(κ)to derive conditions for the discreteness of the imageΠωof perω.In parituclar, we describe examples in whichΠωis not discrete.For the case where K is ﬁnite-dimensional,our results provide complete information,based on a de-tailed analysis of per C(κu)for the universal invariant formκu in Appendix B.In Section3,we turn to the integrability problem for a general compact manifold M.Our strategy is to compose with pullback homomorphisms γ∗:ΓK→Γ(γ∗K),whereγ:S1→M is a smooth loop,and to determine un-der which conditions the period homomorphism of the corresponding twisted4loop groupΓ(γ∗K)only depends on the homotopy class ofγ.If this condition is not satisﬁed,then our examples show that the period groups cannot be controlled in a reasonable way.Fortunately,the latter condition is equivalent to the following requirement on the curvature R(θ)of the principal connec-tion1-formθcorresponding to∇and the action L(ρk):For each derivation D∈im( L(ρk)◦R(θ)),the periods of the2-cocycleηD(x,y):=κ(x,D.y) have to vanish.This condition is formulated completely in terms of K and it is always satisﬁed if K isﬁnite-dimensional becauseπ2(K)vanishes in this case.If the curvature requirement is fulﬁlled,thenΠωis contained in H1dR(M,V),so that we can use integration maps H1dR(M,V)→V to reduce the discreteness problem forΠωto bundles over S1.The second part of Section3treats the lifting problem for the important special case of gauge bundles K=Ad(P)andΓK=Gau(P).In this case we even show that the action of the full automorphism group Aut(P)on ΓK=gau(P)lifts to an action on the central extensiong au(P),deﬁned by ω.We also give an integrability criterion of this action to central extensions of the identity component Gau(P)0.Summarizing,we obtain for gauge bundles the following theorem:Theorem0.1Ifπ0(K)isﬁnite and dim K<∞,then the following are equivalent:(1)ωκintegrates for each principal K-bundle P over a compact manifold Mto a Lie group extension of Gau(P)0.(2)ωκintegrates for the trivial K-bundle P=S1×K over M=S1to a Liegroup extension of C∞(S1,K)0.:π3(K)→V is discrete.(3)The image of perωκThese conditions are satisﬁed ifκis the universal invariant symmetric bilin-ear form with values in V(k).In order to increase the readability of the paper,we present some back-ground material in appendices.This comprises the Lie group structure on groups of sections of Lie group bundles,a discussion of the universal invari-ant form forﬁnite-dimensional Lie algebras,the main results on integrating Lie algebra extensions to Lie group extensions and some curvature issues for principal bundles,needed in Section3.5Notation and basic conceptsA Lie group G is a group equipped with a smooth manifold structure modeled on a locally convex space for which the group multiplication and the inversion are smooth maps(cf.[Mil84],[Ne06]and[GN08]).We write1∈G for the identity element andλg(x)=gx,resp.,ρg(x)=xg for the left,resp.,right multiplication on G.Then each x∈T1(G)corresponds to a unique left invariant vectorﬁeld x l with x l(g):=dλg(1).x,g∈G.The space of left invariant vectorﬁelds is closed under the Lie bracket of vectorﬁelds,hence inherits a Lie algebra structure.In this sense we obtain on T1(G)a continuous Lie bracket which is uniquely determined by[x,y]l=[x l,y l]for x,y∈T1(G). We write L(G)=g for the so obtained locally convex Lie algebra and note that for morphismsϕ:G→H of Lie groups we obtain with L(ϕ):=T1(ϕ) a functor from the category of Lie groups to the category of locally convex Lie algebras.We write q G: G0→G0for the universal covering map of the identity component G0of G and identify the discrete central subgroup ker q G of G0withπ1(G)∼=π1(G0).For a smooth map f:M→G we deﬁne the(left)logarithmic derivative inΩ1(M,g)byδ(f)v m:=f(m)−1·T m(f)v m,where·refers to the two-sided action of G on its tangent bundle T G.In the following we always write I=[0,1]for the unit interval in R.A Lie group G is called regular if for eachξ∈C∞(I,g),the initial value problem γ(0)=1,γ′(t)=γ(t)·ξ(t)=T(λγ(t))ξ(t)has a solutionγξ∈C∞(I,G),and the evolution mapevol G:C∞(I,g)→G,ξ→γξ(1)is smooth(cf.[Mil84]).For a locally convex space E,the regularity of the Lie group(E,+)is equivalent to the Mackey completeness of E,i.e.,to the existence of integrals of smooth curvesγ:I→E.We also recall that for each regular Lie group G,its Lie algebra g is Mackey complete and that all Banach–Lie groups are regular([GN08]).A smooth map exp G:g→G is said to be an exponential function if for each x∈g,the curveγx(t):=exp G(tx)is a homomorphism R→G with γ′x(0)=x.Presently,all known Lie groups modelled on complete locally convex spaces possess an exponential function.For Banach–Lie groups,its existence follows from the theory of ordinary diﬀerential equations in Banach6spaces.A Lie group G is called locally exponential,if it has an exponential function mapping an open0-neighborhood in g diﬀeomorphically onto an open neighborhood of1in G.For more details,we refer to Milnor’s lecture notes[Mil84],the survey[Ne06]or the forthcoming monograph[GN08].If q:E→B is a smoothﬁber bundle,then we writeΓE for its space of smooth sections.If g is a topological Lie algebra and V a topological g-module,we write (C•(g,V),d g)for the corresponding Lie algebra complex([ChE48]).1Central extensions of section algebras of Lie algebra bundlesWe now turn to the details and introduce our notation.We write P(M,H,q P) for an H-principal bundle over the smooth manifold M with structure group H and bundle projection q P:P→M.To any such bundle P and to any smooth actionρk:H→Aut(k),we associate the Lie algebra bun-dle K,which is the set(P×k)/H of H-orbits in P×k for the action h.(p,x)=(p.h−1,ρk(h)x).We write[(p,x)]:=H.(p,x)for the elements of K and q K:K→M,[(p,x)]→q P(p)for the bundle projection.It is no loss of generality to assume that the bundle P is connected. Indeed,if P1⊆P is a connected component,then q(P1)=M and H1:= {h∈H:P1.h=P1}is an open subgroup,so that P1is a principal H1-bundle over M.Further,the canonical map P1×k→K,(p,x)→[(p,x)]is surjective and induces a diﬀeomorphism(P1×k)/H1∼=K.In the following we shall always assume that P is connected.This implies that the connecting map δ1:π1(M)→π0(H)of the long exact homotopy sequence of P is surjective.Further,let V be a Fr´e chet H-module on which the identity component H0 acts trivially andρV:H→GL(V)the corresponding representation,so that H0⊆kerρV and we obtain an induced representationρV to the squeezed bundle P0:= P/H0,which is aπ0(H)-principal bundle over M.Due to theﬂatness of V,we have a natural exterior derivative d on the spaceΩ•(M,V)∼=Ω•(P0,V)π0(H) of V-valued diﬀerential forms and we deﬁneinherits a natural Hausdorﬀlocally convex topology.In fact,in Lemma3.8 below we construct a continuous mapΩ1(M,V)→Z1(π1(M),V)(group cocycles with respect to the representationρM:=ρV◦δ1)and show that dΓV is the inverse image of the space B1(π1(M),V)of coboundaries which is ﬁnite-dimensional if V is so,hence closed in the Fr´e chet space Z1(π1(M),V). Therefore dΓV is closed.IfρV(H)=ρM(π1(M))isﬁnite,then M:= M/kerρM is aﬁnite covering manifold of M and D:=π1(M)/kerρM acts by deck transformations.We then haveΩ•(M,V)∼=Ω•( M,V)D and theﬁniteness of D implies that dΓV∼= B1dR( M,V)D,so that dΓV is a closed subspace.We therefore assume in the following that eitherρV(H)isﬁnite or that V isﬁnite-dimensional to ensure thatwhereθ∈Ω1(P,h)is the principal connection1-form corresponding to the connection∇.Proposition1.1The prescriptionω(f,g):=ω∇κ(f,g):=[κ(f,d∇g)]deﬁnes a Lie algebra cocycle onΓK with values in the trivialΓK-module Remark1.2Since the spaceas we see with similar calculations as in the proof Proposition1.1.For the trivial k-module V,we write Sym2(k,V)k for the space of V-valued symmetric invariant bilinear forms,and recall the Cartan mapC:Sym2(k,V)k→Z3(k,V),C(κ)(x,y,z):=κ([x,y],z).We say thatκis exact if C(κ)is a coboundary.If C(κ)=d kηfor some η∈C2(k,V),thenη)(f,g,h))=− cyc.d(η([f,g],h)),d(κ([f,g],h))=d((dkso thatωκ,η(f,g):=κ(f,d∇g)−κ(g,d∇f)−d(η(f,g))is anΩ1(M,V)-valued2-cocycle onΓK lifting2ω∇κ(cf.[Ne08,Sect.2]). Remark1.3Ifβ∈Ω1(M,Ad(P))is a bundle-valued1-form,then we obtain for each X∈V(M)a derivationβk(X):=ρk◦β(X)ofΓK and this derivation preserves the symmetric bilinearΓV-valued map(f,g)→κ(f,g),so thatηβ(f,g):=κ(f,β.g)deﬁnes anΩ1(M,V)-valued2-cocycle onΓK.For∇′=∇+β,we now haveω′−ω=qΩ◦ηβ,where qΩ:Ω1(M,V)→Furthermore,the spaceΓV carries a natural V(M)-module structure which we may pull back to an aut(P)-module structure for which the ideal gau(P) acts trivially.Since L(ρk)=ad,Proposition1.1implies that in this situation the coho-mology class[ω]=[ω∇κ]∈H2(gau(P),Ω1(M,V),and a necessary condition for the existence of a Lie group extension integrat-ingωis that the imageΠωof the period map,the period group,is discrete ([Ne02a],Theorem VII.9).To obtain information on this period group,our strategy isﬁrst to take a closer look at the case M=S1and then to use this case to treat more general situations.The much simpler case of trivial bundles has been treated in a similar fashion in[MN03].11A class of examplesExample1.6Letπ:Q→M be a compact locally trivial smooth bundle with(compact)ﬁber N.Then Q is associated to the H:=Diﬀ(N)-principal bundle P withﬁber P m:=Diﬀ(N,Q m)with the canonical H action by composition.For any locally convex Lie group G,we have a canonical H-action on K:=C∞(N,G)by(ϕ,γ)→γ◦ϕ−1whose smoothness follows from the smoothness of the action of Diﬀ(N)on N and the smoothness of the evaluation map of K(cf.[NeWa07],Lemma A.2)and we thus obtain an associated Lie group bundle K:=P×H C∞(N,G).The sections of the Lie group bundleΓK may be identiﬁed with the set C∞(P,K)H of H-equivariant smooth functions P→K.Proposition1.7If G is locally exponential,then the maps:C∞(Q,G)→ΓK,s f(p)=f◦p,p∈P m=Diﬀ(N,P m)is an isomorphism of Lie groups.Proof.In local coordinates one easily checks that s actually is an isomor-phism of abstract groups compatible with the smooth compact open topology, so that it actually is an isomorphism of topological groups.If,in addition,G is assumed to be locally exponential,thenΓK and C∞(Q,G)inherit this property(Theorem A.1),and now the general the-ory of locally exponential Lie groups([GN08])implies that the topological isomorphism between these groups actually is a diﬀeomorphism,hence an isomorphism of Lie groups.2Lie group bundles over the circleThroughout this section we consider the special case M=S1and assume that the Lie group K is regular.Then every K-Lie group bundle over S1 isﬂat,hence determined by its holonomyϕ∈Aut(K).Conversely,every automorphismϕ∈Aut(K)leads to a Lie group bundle Kϕ=R×ϕK over S1with holonomyϕ.Indeed,Kϕis the Lie group bundle associated to the universal covering q S1:R→S1∼=R/Z by the action of H:=Z∼=π1(S1)on K deﬁned byϕ.The smooth sections of Kϕcorrespond to twisted loops:ΓKϕ∼=C∞(R,K)ϕ:={f∈C∞(R,K):(∀t∈R)f(t+1)=ϕ−1(f(t))}. From now on we identifyΓK with C∞(R,K)ϕand writeev K0:C∞(R,K)ϕ→K,f→f(0),for the evaluation homomorphism in0.On the Lie algebra level,we similarly get withϕk= L(ϕ)a Lie algebra bundle KϕkwithΓKϕk ∼=C∞(R,k)ϕk:={f∈C∞(R,k):(∀t∈R)f(t+1)=ϕ−1k(f(t))}and L(ev K0)=ev k0.2.1On the topology of twisted loop groupsIn Section3we shall reduce the calculation of the period groups of forω∇κessentially to the case M=S1,so that we need detailed information on the second homotopy group of twisted loop groups.A central tool is a simple description of the connecting maps in the long exact homotopy sequence deﬁned by the evaluation map ev K0for a twisted loop group C∞(R,K)ϕ, which is based on the fact that the passage from smooth to continuous twisted loops is a weak homotopy equivalence.Lemma2.1The image of the evaluation homomorphism ev K0is the opensubgroupK[ϕ]:={k∈K:ϕ(k)k−1∈K0}={k∈K:kK0∈π0(K)ϕ}. Proof.For f∈C∞(R,K)ϕ,we have f(1)=ϕ−1(f(0))∈f(0)K0,so that the image of ev K0is contained in K[ϕ].If,conversely,k∈K[ϕ],then there exists a smooth curveα:[0,1]→K withα(0)=k,α(1)=ϕ−1(k)such thatαis locally constant near0and1.Then f(n+t):=ϕ−n(α(t))for t∈[0,1],n∈Z,deﬁnes a section of Kϕwith f(0)=k.Lemma2.2The Lie group homomorphism ev K0has smooth local sections,hence deﬁnes a Lie group extension of K[ϕ]by C∞∗(R,K)ϕ=ker ev K0. Proof.That ev K0has smooth local sections can be seen as follows.Let (ψ,U)be a chart of K,centered in1for whichψ(U)is convex.Let further h:[0,1]→R be a smooth function with h(0)=0and h(1)=1which is constant in[0,ε]and[1−ε,1].Let W⊆U∩ϕ(U)be a1-neighborhood in K.For k∈W we then consider the smooth curveγk:[0,1]→K,γk(t):=ψ−1 (1−h(t))ψ(k)+h(t)ψ(ϕ−1(k)) . Thenγk is constant near0and1,γk(0)=k,andγk(1)=ϕ−1(k).We extend γk smoothly to R in such a way that it deﬁnes an element on C∞(R,K)ϕ. Then the smoothness of the map W→C∞(R,K)ϕ,k→γk follows from the smoothness of the corresponding map W×R→K,(k,t)→γk(t),which in turn follows from its smoothness on each subset W×]n−ε,n+1+ε[(cf. [GN08]).Corollary2.4The inclusion C∞∗(R,K)ϕ֒→C∗(R,K)ϕof the smooth based twisted loop group into the continuous based twisted loop group is a weak homotopy equivalence.14Proof.In view of Lemma2.2,the evaluation ev K0deﬁnes a smoothlylocally trivialﬁber bundle C∞(R,K)ϕ→K[ϕ],and a similar(even simpler) argument shows that the same holds for the continuous twisted loop group. Since id K[ϕ]and the inclusion C∞(R,K)ϕ֒→C(R,K)ϕare weak homotopy equivalences,the5-Lemma implies that the same holds for the inclusion C∞∗(R,K)ϕ֒→C∗(R,K)ϕof theﬁbers(cf.Prop.A.8in[Ne02c]).Remark2.6(a)We have a short exact sequence1→π1(K)ϕ:=π1(K)/im(π1(ϕ)−id)֒→π0(C∞(R,K)ϕ)→→π0(K)ϕ→1. If K is connected,we obtain in particularπ0(C∞(R,K)ϕ)∼=π1(K)ϕ.(b)For the evaluation of period maps,important information is contained in the short exact sequence1→π3(K)ϕ֒→π2(C∞(R,K)ϕ)→→π2(K)ϕ→1.Ifπ2(K)vanishes,it follows that the corresponding mapπ3(K)∼=π2(ΩK)→π2(C∞(R,K)ϕ)is surjective.15Example2.7We discuss some examples whereϕacts non-trivially onπ2(K).(a)A typical example of a Lie group K for whichπ2(K)is non-trivial is the projective unitary group PU(H)of an inﬁnite-dimensional complex Hilbert space H([Ku65]).Each automorphism of this simply connected group either is induced by a unitary or an anti-unitary map.In fact,the simple connectedness of PU(H)implies Aut(PU(H))∼=Aut(pu(H)),and since the Lie algebra u(H)is the universal central extension of pu(H)([Ne02b, Example III.6]),each automorphism of pu(H)lifts to a unique automorphism of u(H),so thatAut(pu(H))∼=Aut(u(H))∼=Aut∗(gl(H))∼=PU(H)⋊Z/2,where the latter isomorphism follows from Prop.3in Section II.13of[dlH72] and Aut∗denote the group of all automorphismϕwithϕ(x∗)=ϕ(x)∗.We conclude thatπ0(Aut(PU(H)))∼=Z/2.Conjugation with an anti-unitary map induces the inversion on the center T id H of U(H),and this implies that the action ofπ0(Aut(PU(H)))induces the inversion onπ1(Z(U(H)))∼=π2(PU(H)).(b)Another example is the smooth loop group C∞(S1,C)of a compact simple simply connected Lie group C which satisﬁesπ2(C∞(S1,C))∼=π3(C)∼=Z.Its automorphism group isAut(C∞(S1,C))∼=C∞(S1,Aut(C))⋊Diﬀ(S1)(cf.[PS86,Prop.3.4.2])whose group of connected components isπ1(Aut(C))⋊π0(Aut(C)) ⋊Z/2∼= Z(C)⋊π0(Aut(C)) ⋊Z/2,henceﬁnite.In particular,any orientation reversing diﬀeomorphism acts on π2(C∞(S1,C))∼=Z by inversion.2.2Period maps for twisted loop groupsLetκ:k×k→V be a k-invariant symmetric bilinear form andϕV∈GL(V) (deﬁning a Z-module structure on V)withϕV(κ(x,y))=κ(ϕk.x,ϕk.y)for x,y∈k.16We write V:=VϕV for the vector bundle over S1withﬁber V and holonomyϕV.Then the cocycleω∇κcorresponding to the canonical connection∇on Kϕdeﬁned by d∇f=f′(t)dt is given byωϕ(f,g):=[κ(f,d∇g)]= 10κ(f,g′)dt ∈VϕV∼=Ω1(S1,V)∼=H1dR(S1,V)→VϕV=coker(ϕV−id V),[f·dt]→ 10f(t)dtfor f∈ΓV∼=C∞(R,V)ϕVis a linear isomorphism.Proof.Write ddtimplies that evaluation in dThis proves injectivity.To obtain surjectivity,choose someγ:[0,1]→[0,1] which is smooth,constant on a neighborhood of the boundary and satisﬁes γ(0)=0,γ(1)=1and 10γ(t)dt=1Remark2.9If V is inﬁnite-dimensional,we further assume that the image of the operatorϕV−id V is closed,so thatper C(κ):π3(K)→V.2More generally,ifσ:S2→C∞(R,K)ϕis a smooth map and σ:R×S2→K deﬁned by σ(t,m):=σ(m)(t),thenperωϕ([σ])=12 10κ(f,g′)−κ(g,f′)dt satisﬁesσ ωlϕ=12[δ σ,δ σ]=0,the derivative of this curve can be calculated by evaluating it on some smooth vectorﬁeld X∈V(S2):(δσ)′(X)=L∂∂tδ σ (X)= i∂∂tδ σ (X)= −1∂t[δ σ,δ σ]+dδ( σ) (X)=−[δ σ(∂Using the Maurer–Cartan equation forδσ,we further get in C∞(R,Ω2(S1,k)):δσ∧κ(δσ)′=δσ∧κdδ( σ)+δσ∧κ[δσ,δ σ]=δσ∧κdδ( σ)+[δσ,δσ]∧κδ σ=−d(δσ∧κδ σ +d(δσ)∧κδ σ+[δσ,δσ]∧κδ σ=−d(δσ∧κδ σ −12[δσ,δσ]∧κδ σ=−d(δσ∧κδ σ +C(κ)(δσ,δσ,δ σ)and henceσ ωlϕ= S2 ωϕ(δσ,δσ)=12 10S2C(κ)(δσ,δσ,δ σ)dt=1∂t,v,w)for v,w∈T m(S2).4tr(ad x ad y)for the normalized invariant bilinear form,satisfyingκ(x,x)=2 x 2for each x∈su2(C)=span R{I,J,K}.For the basis elements I,J,K,we then haveκ([I,J],K)=2κ(K,K)=4,so that the left invariant3-form deﬁned by C(κ)(x,y,z):=κ([x,y],z)on SU2(C)∼=S3 is4µS3,whereµS3is the volume form of S3.It follows in particular,that per C(κ)l([id K])= SU2(C)C(κ)l=4vol(S3)=8π2.(7)20On the other hand,it has been shown in[PS86](see also the calculationsin Appendix IIa to Section II in[Ne01])that12πωκ=2πZ forωκ:= ωid,so thatΠωκ=4π2Z.In view of the preceding lemma,this is a direct consequence of(7).Let q V:V→VϕV denote the projection map.Then the cocycle ωϕsatis-ﬁesq V◦ ωϕ= L(ι)∗ωϕ,whereι:C∞∗(R,K)ϕ֒→C∞(R,K)ϕdenotes the inclusion map.Therefore Remark C.2yieldsperωϕ◦π2(ι)=per L(ι)∗ωϕ=q V◦per eωϕ=q V◦per C(κ).(8)Ifπ2(K)vanishes,thenπ2(ι)is surjective(Remark2.6)and we thus obtain: Theorem2.12Ifπ2(K)vanishes,thenΠωϕ⊆q V im(per C(κ)) .As a consequence,we obtain forﬁnite dimensional groups with Theo-rem B.11and Cartan’s Theorem(Remark B.8).Corollary2.13If K isﬁnite-dimensional,V=V(k)andκ=κu is univer-sal,then the period groupΠωϕis discrete.We now present an example where the period group depends signiﬁcantly on the connection∇.Example2.14We consider the special case of Remark1.8,whereπ:Q=T2→M=T=R/Z,π(t,s)=s,H=T and K=C∞(T,G)for a compact simple Lie group G.ThenΓK∼= C∞(T2,G)and K∼=T×K is a trivial bundle.To a positive deﬁnite invariant bilinear formκg on g,we associate the invariant bilinear formκ(f,g):= 10κg(f(t),g(t))dt.21The group H=T acts on K,resp.,k,by composition,and the action of the Lie algebra h∼=R is given by D.f=f′,which leavesκinvariant.The Lie algebra cocycleηD(f,g):=κ(f,D.g)= 10κg(f(t),g′(t))dt,is a universal cocycle of k(cf.[PS86]).In particular,the period homomor-phismperηD =1∂s+h(s)·∂f∂s)dsω0(f,g):=+ 10h(s)κg(f,∂g∂s)ds dt,η(f,g)= 10 10h(s)κg(f,∂g∂t)d t+κg(f,∂gΩ1(T2,R).Ifκg is suitably normalized,the period group of this cocycle isΠωu=Z[d t]+Z[d s]([MN03]).We further ﬁnd with Remark B.8π2(C∞(T2,G))∼=π2(G)⊕π3(G)2⊕π4(G)∼=Z2⊕π4(G),22and in these terms(m,n,u)=m[d t]+n[d s].perωuIfγt(s)=γs(t)=(t,s)describes the vertical and horizontal circles in T2, thenω0(f,g)= 10Iγt◦ωu(f,g)dt,so that the period map is given byperω(m,n,u)= 10Iγt◦perωu(m,n,u)dt= 10n dt=n.Similarly,η(f,g)= 10h(s)Iγs◦ωu(f,g)ds,and its period map isperη(m,n,u)= 10h(s)Iγs perωu(m,n,u)ds= 10h(s)m ds=m 10h(s)ds. This implies that the period groupΠω=Z+Z· 10h(s)dsis discrete if and only if the integral 10h(s)ds is rational.Remark2.15We have seen in Remark2.6that for a twisted loop group LϕK:=C∞(R,K)ϕ,ϕ∈Aut(K),the groupπ2(LϕK)is determined by a short exact sequence1→π3(K)ϕ→π2(LϕK)→π2(K)ϕ→1.can be determined in a two-step process. Accordingly,the period groupΠωϕThe restriction toπ3(K)ϕis,up to the factor1/ΠC(κ),ϕperω:π2(K)ϕ→VϕVwhose image determines the period groupΠωas its inverse image in Vϕ.V23The following example shows that both partsπ3(K)ϕandπ2(K)ϕmay contribute non-trivially toΠωand that the period group depends seriously onϕ.Example2.16(a)Let G be a simply connected simple compact Lie group and K:=C∞(S1,G)its loop group.LetϕK=(h,ψ)∈Aut(K)∼=C∞(S1,Aut(G))⋊Diﬀ(S1)(cf.[PS86,Prop.3.4.2]).Here Aut(K)actually carries a natural Lie group structure and the automorphismϕV of V=R induced byϕK for which the Cartan–Killing formκis invariant is±id V,depending on whetherψpreserves the orientation of S1or not.We also note thatπ0(C∞(S1,Aut G))∼=π1(Aut G)∼=Z(G)acts trivially on all higher homotopy groups of G,hence in particular on π3(G)∼=π2(K).ForϕV=−id V we obtain in particular VϕV ={0},so that all periodsvanish.In the latter case,the natural identiﬁcation ofπ2(K)withπ3(G) shows thatϕK acts as−id onπ2(K)∼=π3(G)∼=Z,so thatπ2(K)ϕ={0}.IfϕV=id V,then VϕV =V=R andψis orientation preserving.Thenthe action ofϕK onπ2(K)∼=π3(G)is trivial.The action ofϕK onπ3(K)∼=π3(G)×π4(G)is trivial on theﬁrst factor(coming from constant functions) andπ4(G)isﬁnite,so thatπ3(K)ϕis of rank1.Now the same arguments as in Example2.14show thatπ2(LϕK)is of rank2and both summands contribute toΠω.3Corresponding Lie group extensions We now determine in which cases the central extension ΓK deﬁned by the cocycleω∇κintegrates to a Lie group extension.To this end we analyze its period groupΠω:=im(perω∇κ)and determine whether the adjoint action of ΓK lifts to an action on ΓK(cf.Appendix C and[Ne02a]).Throughout,M denotes a compact connected manifold.24。

1101.4484v1 Some general results on conformal embeddings of affine vertex operator algebras

a r X i v :1101.4484v1[mat h.Q A]24J a n211SOME GENERAL RESULTS ON CONFORMAL EMBEDDINGS OF AFFINE VERTEX OPERATOR ALGEBRAS DRA ˇZEN ADAMOVI ´C AND OZREN PER ˇSE Abstract.We give a general criterion for conformal embeddings of vertex operator algebras associated to aﬃne Lie algebras at arbitrary ing that criterion,we construct new conformal embeddings at admissible rational and negative integer levels.In particular,we con-struct all remaining conformal embeddings associated to automorphisms of Dynkin diagrams of simple Lie algebras.The semisimplicity of the corresponding decompositions is obtained by using the concept of fusion rules for vertex operator algebras.1.Introduction Let U and V be vertex operator algebras of aﬃne type.We say that U is conformally embedded into V if U can be realized as a vertex subalge-bra of V with the same Sugawara Virasoro vector.The most interesting case of conformal embedding is when V is a ﬁnite direct sum of irreducible U –modules.These conformal embeddings are studied in various aspects of conformal ﬁeld theory (cf.[6],[8],[9],[15],[22],[36]),the theory of tensor cat-egories (cf.[21],[28])and in the representation theory of aﬃne Kac-Moody Lie algebras (cf.[12]).The construction and classiﬁcation of conformal embeddings have mostly been studied for simple aﬃne vertex operator alge-bras of positive levels.Some examples of conformal embeddings at rational admissible levels were constructed in [33]and [35].In the present paper we give both necessary and suﬃcient conditions forconformal embeddings at general levels,within the framework of vertex op-erator algebra theory.In this way we will be able to construct new examples of conformal embeddings at rational and negative integer levels.Let us ex-plain our results in more details.Let g be a simple ﬁnite-dimensional Lie algebra and g 0its subalgebra which is a reductive Lie algebra.Let N g (k,0)be the universal aﬃne vertex operator algebra of level k =−h ∨associated to g ,and L g (k,0)its simple quotient.Then g 0generates a subalgebra U g0(resp. U g 0)of N g (k,0)(resp.L g (k,0)).Let ωg 0be the Virasoro vector in U g 0which is a sum of usualVirasoro vectors obtained by using Sugawara construction.Let 2000Mathematics Subject Classiﬁcation.Primary 17B69,Secondary 17B67,81R10.12DRAˇZEN ADAMOVI´C AND OZREN PERˇSEn∈ZL(0)x(−1)1=x(−1)1(x∈W).(1.1)Moreover,the condition(1.1)implies that N g(k,0)contains a singular vector of conformal weight2.If k is a positive integer,then N g(k,0)has a singular vector of conformal weight2if and only if k=1.So we only have conformal embeddings into L g(1,0).But in general we have such singular vectors for other values of k (cf.Remark2.4).Next we apply Theorem1.1in the case when g0is a simple Lie algebra, andg=g0⊕V g0(µ1)⊕···⊕V g(µs),where V g0(µi)is irreducibleﬁnite-dimensional g0-module with highest weightµi,and V g(µi)(i=1,...,s)is orthogonal to g0with respect to the invariant bilinear form on g.Denote by L g0(k′,0)the subalgebra of L g(k,0)generated by g0,where k′=ak and a∈N is the Dynkin index of the embedding g0<g.Note that in the case of general levels,vertex algebra L g0(k′,0)is not necessarily simple.Theorem1.1then implies that the key condition for the conformal em-bedding is that the eigenvalue of the Casimir operator of g0is the same for all V g(µi),i=1,...,s.More precisely,if(µi,µi+2ρ0)0=(µj,µj+2ρ0)0for all i,j=1,...,s, where(·,·)0denotes the suitably normalized invariant bilinear form on g0, then for k,k′such that(µi,µi+2ρ0)03 Our methods enable us to construct all conformal embeddings associated to automorphisms of Dynkin diagrams.Let us illustrate this result by the following table:Table A,g,g0are simple Lie algebras,g0<g.g decomposition of L g(k,0)k′Cℓ−1A2ℓL g0(k′,0)⊕L g(k′,2ω1)2A11DℓL g0(k′,0)⊕L g(k′,ω1)−ℓ+2F4−3D4L g0(k′,0)⊕L g(k′,ω2)⊕−2L(1)g0(k′,ω1)⊕L(2)g(k′,ω1)In order to describe all conformal embeddings for such automorphisms,we include into this table some well-known conformal embeddings at positive integer levels in the cases(A2ℓ,Bℓ),(A2,A1)(cf.[22];see also[4],[37])and the conformal embedding for pair(A2ℓ−1,Cℓ)from[5].But other conformal embeddings at negative integer levels are new.We also apply our methods to aﬃne vertex operator algebras at admissible levels(cf.[1],[3],[26],[27],[32],[33],[34],[37])and get new conformalembedding of L A2(−5/3,0)into L G2(−5/3,0)with decompositionL G2(−5/3,0)=L A2(−5/3,0)⊕L A2(−5/3,ω1)⊕L A2(−5/3,ω2).We also determine the decomposition for the conformal embedding of L Dℓ(−ℓ+3/2,0)into L Bℓ(−ℓ+3/2,0)from[35].We should also mention that our methods give an uniform proof for a family of conformal embeddings which does not require explicit realizations of aﬃne Lie algebras or explicit formulas for singular vectors.In most cases one can check all conditions for conformal embeddings only by using tensor product decompositions ofﬁnite-dimensional modules of simple Lie algebras. The notion of conformal embedding is also closely related to the notion of extension of vertex operator algebra.We note that simple current extensions of aﬃne vertex operator algebras at positive integer levels were studied in [13],[31].We assume that the reader is familiar with the notion of vertex operator algebra(cf.[10],[18],[19]).2.A general criterion for conformal embeddingsIn this section we shall derive a general criterion for the conformal embed-ding of aﬃne vertex operator algebras associated to a pair of Lie algebras (g,g0),where g is a simple Lie algebra and g0its reductive subalgebra.Let g be the simple complexﬁnite-dimensional Lie algebra with a non-degenerate invariant symmetric bilinear form(·,·).We can normalize the form on g such that(θ,θ)=2,whereθis the highest root of g.Letρbe the4DRAˇZEN ADAMOVI´C AND OZREN PERˇSEsum of all fundamental weights for g and let h∨be the dual Coxeter number for g.Assume that g0is a Lie subalgebra of g such that(1)bilinear form(·,·)on g0is non-degenerate and g=g0⊕W is anorthogonal sum of g0and a g0–module W;(2)g0=⊕n i=0g0,i is an orthogonal sum of commutative subalgebra g0,0and simple Lie algebras g0,1,...,g0,n(i.e.,g0is a reductive Lie alge-bra).Let(·,·)g0,i be the non-degenerate bilinear form on g0,i normalized suchthat(θ0,i,θ0,i)g0,i =2,whereθ0,i is the highest root of g0,i.One can showthata i(·,·)g0,i=(·,·)|g0,i×g0,i for certain a i∈N.(For i=0,we can take a0=1).The(n+1)–tuple(a0,a1,···,a n)is called the Dynkin multi-index of embedding g0into g.Let{u i,j}j,{v i,j}j be a pair of dual bases for g0,i such that(u i,j,v i,k)g0,i=δj,k.Let h∨0,i be the dual Coxeter number for g0,i.(For i=0we take h∨0,0=0).Let N g(k,0)be the universal aﬃne vertex operator algebra of level k=−h∨associated to g(cf.[17],[20],[25],[29],[30]).Let N1g(k,0)be the max-imal ideal of N g(k,0),and L g(k,0)=N g(k,0)/N1g(k,0)the corresponding simple vertex operator algebra.Then g0,i generates a vertex subalgebra of N g(k,0)which is isomorphicto N g0,i (k i,0)for k i=a i k.Moreover,g0generates a subalgebra of N g(k,0)which is isomorphic toU g0=⊗n i=0N g0,i(k i,0).Let U g0be the subalgebra of L g(k,0)generated by g0.Then U g0is iso-morphic toU g0=⊗n i=0 L g0,i(k i,0), where L g0,i(k i,0)is a certain quotient of N g0,i(k i,0).Deﬁnition2.1.We say that the vertex operator algebra U g0is conformally embedded into L g(k,0),if U g0is a vertex subalgebra of L g(k,0)with the same Virasoro vector.Letωg be the usual Sugawara Virasoro vector in N g(k,0)and letωg0,i =15 As usual we identify x∈g with x(−1)1∈N g(k,0).Therefore,W can be considered as a subspace of N g(k,0).LetL(n)z−n−2=Y(ωg,z).Theorem2.2.Assume that2(k i+h∨0,i)[[x,u i,j],v i,j](−1)1=k1+h∨0,1+···+k n dim(g0,1)k+h∨.(2.6)So our condition(2.1)from Theorem 2.2implies numerical criterion (2.6).But,in general,(2.6)is only a necessary condition for conformal embeddings.Remark2.4.Assume that the conditions from Theorem2.2are satisﬁed. Then N g(k,0)must have at least one singular vector at conformal weight2. Some explicit formulas for these singular vectors appeared in[1],[2],[32], [33],[34]and recently in[7].We shall now see how the theory presented hereﬁts into some examples.6DRA ˇZEN ADAMOVI ´CAND OZREN PER ˇSE Example 2.5.Let g be a simple complex Lie algebra of type A ,D or E ,h its Cartan subalgebra and ∆its root system.Then g =h ⊕W,W = α∈∆g α.Take now g 0=h in the above construction.ThenL (0)|W ≡Id iﬀk =1or k =−1/2.This leads to the conformal embeddings L g 0(2,0)<L g (1,0), L g 0(−1,0)<L g (−1/2,0).Example 2.7.By using the same method as above we obtain the following conformal embeddings:(1)Let g =sl (ℓ+1,C ),g 0=gl (ℓ,C ).Then we have conformal embeddings:L g 0(1,0)<L g (1,0),L g 0(−ℓ+12,0).(2)Let g =o (2ℓ,C ),g 0=gl (ℓ,C ).Then we have conformal embeddings:L g 0(1,0)<L g (1,0), L g 0(−2,0)<L g (−2,0).Remark 2.8.One can show that in examples presented above the vertex operator algebra L g (k,0)is not a ﬁnite sum of irreducible L g 0(k ′,0)–modules.But it is still possible that there exists certain twisted L g (k,0)–module which is a ﬁnite sum of irreducible twisted L g 0(k ′,0)–modules.In particular,this holds for Z 2–twisted module for L g (1,0)which gives principal realization of level one modules for aﬃne Lie algebras in Example 2.5.It seems that to get conformal embeddings such that L g (k,0)is a ﬁnite sum of irreducible L g 0(k ′,0)–modules,one needs to consider the case when g 0is semisimple.In this article we shall make the ﬁrst step and consider the important case when g 0is a simple Lie algebra.So we shall now assume that g 0is a simple Lie algebra.In the above settings we take n =1,g 0=g 0,1.Set(·,·)0=(·,·)g 0,1,h ∨0=h ∨0,1,a =a 1.Let ρ0be the sum of all dominant integral weights for g 0.7 Assume thatg=g0⊕g1⊕···⊕g s,such that(1)g i=V g0(µi)is irreducibleﬁnite-dimensional highest weight g0-modulewith highest weightµi,(2)g i⊥g0for i=1,...,s(with respect to(·,·)),(3)(µi,µi+2ρ0)0=(µj,µj+2ρ0)0(i,j>0).Then g0generates a subalgebra of N g(k,0)isomorphic to U g0∼=Ng0(k′,0),where k′=ak.Letωg(resp.ωg0)be the Virasoro vector in N g(k,0)(resp.N g(k′,0))obtained by the Sugawara construction.In what follows we choose k such that(µi,µi+2ρ0)0L(0)x(−1)1=x(−1)1(x∈W).As before,let U g0= L g0(k′,0)be the subalgebra of L g(k,0)generated by g0.Now Theorem2.2implies the following result: Corollary2.9.Assume that the above conditions hold.Then L g0(k′,0)is conformally embedded into L g(k,0).The simplicity of the vertex operator algebra L g0(k′,0)will be investigated in the next section.3.On semisimplicity of L g(k,0)as g0–moduleIn this section we give suﬃcient conditions for complete reducibility of L g(k,0)as g0–module.In particular,we obtain the conformal embedding ofsimple vertex operator algebras L g0(k′,0)<L g(k,0).We assume that g0is a simple Lie algebra and that all conditions on embedding g0<g from Section2hold.Furthermore,we assume that g0is theﬁxed point subalgebra of an automorphismσof g of order s+1,and thatV g0(µi)is the eigenspace associated to the eigenvalueξi(for i=1,...,s),whereξdenotes the corresponding primitive root of unity.Thenσcan be extended to aﬁnite-order automorphism of the simple vertex operator algebra L g(k,0)which admits the following decompositionL g(k,0)=L g(k,0)0⊕L g(k,0)1⊕···⊕L g(k,0)swhereL g(k,0)i={v∈L g(k,0)|σ(v)=ξi v}.Clearly L g(k,0)i is a g0–module.For a dominant weightµfor g0we deﬁne µ=k′Λ0+µ.Denote by L g(k′,µ)the irreducible highest weight g0–module with highest weight µ.8DRAˇZEN ADAMOVI´C AND OZREN PERˇSENote that the lowest conformal weight of any g0–module with highest weight µis given by the formula(µ,µ+2ρ0)09Table1.g,g0are simple Lie algebras,g0<g.g decomposition of g k′Cℓ−1Dℓg0⊕V g0(ω1)−ℓ+2F4−3G2g0⊕V g0(ω1)⊕V g(ω2)−5/3Dℓ−ℓ+3/210DRAˇZEN ADAMOVI´C AND OZREN PERˇSELemma4.3.We have:(1)The lowest conformal weights of A(1)2–modules with highest weights−113Λ0+Λ1+Λ2are54,respectively.(2)The lowest conformal weights of B(1)ℓ−1–modules with highest weights−ℓΛ0+ 2Λ1and−ℓΛ0+Λ2are2+1ℓ−1,respectively.(3)The lowest conformal weights of D(1)ℓ–modules with highest weights(−ℓ−12)Λ0+Λ2are2+22ℓ−1,respectively.Lemma4.4.As an F(1)4–module,L E6(−3,0)does not contain singular vec-tors of weights−7Λ0+2Λ4,−5Λ0+Λ1and−7Λ0+Λ3.Proof.Weﬁrst note that F(1)4–modules with highest weights−7Λ0+2Λ4and−5Λ0+Λ1do not have integral conformal weights.(The correspondinglowest conformal weights are132,respectively.)The lowest conformal weight of module of highest weight−7Λ0+Λ3forF(1) 4is2,but one can directly check that there is no non-trivial singular vec-tor of that weight in L E6(−3,0).We use the construction of the root systemof type E6from[11],[23].For a subset S={i1,...,i k}⊆{1,2,3,4,5}withodd number of elements(i.e k=1,3or5),denote by e(i1...i k)the(suitablychosen)root vector associated to positive root1115.Conformal embedding of L G 2(−2,0)into L B 3(−2,0)andL D 4(−2,0)In this section we consider the conformal embedding associated to a pair of simple Lie algebras (D 4,G 2).It turns out that this case is more complicated and one can not directly apply general results from Section 3.The main diﬀerence between this case and cases studied in Section 4is that L D 4(−2,0)contains a non-trivial singular vector for G (1)2of conformal weight 2.In order to prove that L D 4(−2,0)is completely reducible L G 2(−2,0)–module we shall need more precise analysis,which uses the conformal embedding of L B 3(−2,0)in L D 4(−2,0)from Section 4.Let g be the simple complex Lie algebra of type D 4.Then g has an order three automorphism θ,induced from the Dynkin diagram automorphism,such that the ﬁxed point subalgebra g 0is isomorphic to the simple Lie algebra of type G 2.We have the decomposition of g into eigenspaces of θ:g =g 0⊕V (1)g 0(ω1)⊕V (2)g 0(ω1),(5.1)where V (1)g 0(ω1)is generated by highest weight vector e ǫ1−ǫ4+ξe ǫ1+ǫ4+ξ2e ǫ2+ǫ3and V (2)g 0(ω1)by e ǫ1−ǫ4+ξe ǫ2+ǫ3+ξ2e ǫ1+ǫ4,for suitably chosen root vectors.(Here ξdenotes the primitive third root of unity.)Lie algebra g 0is also a subalgebra of Lie algebra of type B 3considered in Section 4,which we now denote by g ′.We have g ′=g 0⊕V (3)g 0(ω1),(5.2)where V (3)g 0(ω1)is generated by highest weight vector e ǫ1−ǫ4+e ǫ1+ǫ4−2e ǫ2+ǫ3.By using (5.2),the decomposition of g from Section 4and the fact that V B 3(ω1)∼=V G 2(ω1)as g 0–module,one obtains another decomposition of g :g =g 0⊕V (3)g 0(ω1)⊕V (4)g 0(ω1),(5.3)where V (4)g 0(ω1)is generated by e ǫ1−ǫ4−e ǫ1+ǫ4.By using (5.2)and results from Section 2we see that L G 2(−2,0)is conformally embedded into L B 3(−2,0),which is conformally embedded in L D 4(−2,0).Recall that Theorem 4.1gives that the vertex operator algebra L D 4(−2,0)has an order two automorphism σwhich deﬁnes the following decomposition:L D 4(−2,0)=L B 3(−2,0)⊕L B 3(−2,ω1).(5.4)Denote by L (i )G 2(−2,ω1)the G (1)2–submodule of L D 4(−2,0)generated by top component V (i )g 0(ω1),for i =1,2,3,4.Furthermore,one can directly check that vectorv =(e ǫ1−ǫ3(−1)e ǫ1−ǫ4(−1)+e ǫ2−ǫ4(−1)e ǫ2+ǫ3(−1)+e ǫ2+ǫ4(−1)e ǫ1+ǫ4(−1)−e ǫ1−ǫ3(−1)e ǫ1+ǫ4(−1)−e ǫ2−ǫ4(−1)e ǫ1−ǫ4(−1)−e ǫ2+ǫ4(−1)e ǫ2+ǫ3(−1))112DRA ˇZEN ADAMOVI ´CAND OZREN PER ˇSE is the unique (up to a scalar)non-trivial singular vector of weight −5Λ0+Λ2for G (1)2in L D 4(−2,0).Denote by L G 2(−2,ω2)the G (1)2–submodule gener-ated by v .We also have that θ(v )=v and σ(v )=−v .This implies that v ∈L B 3(−2,ω1).Now we consider L B 3(−2,0)and L B 3(−2,ω1)as G (1)2–modules.We use the following lemmas:Lemma 5.1.We have the following tensor product decompositions:(1)V G 2(ω1)⊗V G 2(ω1)=V G 2(2ω1)⊕V G 2(ω2)⊕V G 2(ω1)⊕V G 2(0),(2)V G 2(ω1)⊗V G 2(ω2)=V G 2(ω1+ω2)⊕V G 2(2ω1)⊕V G 2(ω1).Lemma 5.2.As a G (1)2–module,L D 4(−2,0)does not contain singular vec-tors of weights −6Λ0+2Λ1and −7Λ0+Λ1+Λ2.Proof.The proof follows from the observation that G (1)2–modules of highest weights −6Λ0+2Λ1and −7Λ0+Λ1+Λ2do not have integral conformal weights.(The corresponding lowest conformal weights are 72,respec-tively.) Proposition 5.3.We have:(1)L B 3(−2,0)= L G 2(−2,0)+ L (3)G 2(−2,ω1).(2)L B 3(−2,ω1)= L(4)G 2(−2,ω1)+ L G 2(−2,ω2).Proof.(1)Since v is the unique (up to a scalar)singular vector of weight −5Λ0+Λ2for G (1)2in L D 4(−2,0),it follows that there is no singular vec-tor of that weight in L B 3(−2,0).The tensor product decomposition from Lemma 5.1(1),Lemma 5.2and standard fusion rules arguments now im-ply that L G 2(−2,0)+ L (3)G 2(−2,ω1)is a vertex subalgebra of L B 3(−2,0)which contains generators of L B 3(−2,0).The claim (1)follows.Claim (1),Lemmas 5.1and 5.2and standard fusion rules arguments now implythat L (4)G 2(−2,ω1)+ L G 2(−2,ω2)is L B 3(−2,0)–submodule of the irreducible module L B 3(−2,ω1).This implies claim (2). Theorem 5.4.We have:L D 4(−2,0)=L G 2(−2,0)⊕L G 2(−2,ω2)⊕L (1)G 2(−2,ω1)⊕L (2)G 2(−2,ω1).Proof.First we notice that relation (5.4)and Proposition 5.3give L D 4(−2,0)= L G 2(−2,0)+ L G 2(−2,ω2)+ L (3)G 2(−2,ω1)+ L (4)G 2(−2,ω1),which also impliesL D 4(−2,0)= L G 2(−2,0)⊕ L G 2(−2,ω2)⊕ L (1)G 2(−2,ω1)⊕ L (2)G 2(−2,ω1).By using the fact that Z 3–orbifold components of simple vertex operator algebra are simple (cf.[14]),we get thatV = L G 2(−2,0)⊕ L G 2(−2,ω2)13 is a simple vertex operator algebra,and that L(1)G2(−2,ω1)and L(2)G2(−2,ω1) are its irreducible modules.Since the automorphismσacts as−1on L G2(−2,ω2),it follows that V is Z2–graded,so its graded components are also simple.Theorem3from[14]now implies that L(1)G2(−2,ω1)and L(2)G2(−2,ω1)are inequivalent V–modules,and Theorem6.1from the same paper then implies that L(1)G2(−2,ω1)and L(2)G2(−2,ω1)are irreducible asL G2(−2,0)–modules.The proof follows. It follows from the proof of Theorem5.4that L(3)G2(−2,ω1)and L(4)G2(−2,ω1)are also irreducible L G2(−2,0)–modules,so we obtainCorollary5.5.We have:(1)L B3(−2,0)=L G2(−2,0)⊕L G2(−2,ω1),(2)L B3(−2,ω1)=L G2(−2,ω1)⊕L G2(−2,ω2).References[1] D.Adamovi´c,Some rational vertex algebras,Glas.Mat.Ser.III29(49)(1994),25-40.[2] D.Adamovi´c,A construction of some ideals in aﬃne vertex algebras,Int.J.Math.Math.Sci.(2003),971–980.[3] D.Adamovi´c and as,Vertex operator algebras associated to modular in-variant representations for A(1)1,Math.Res.Lett.2(1995),563–575.[4] D.Adamovi´c and O.Perˇs e,On coset vertex algebras with central charge1,Math.Commun.15(2010),143–157.[5] D.Adamovi´c and O.Perˇs e,The vertex algebra M(1)+and certain aﬃne vertexalgebras of level−1,arXiv:1006.1752.[6]R.C.Arcuri,J.F.Gomes and D.I.Olive,Conformal subalgebras and symmetricspaces,Nuclear Phys.B285(1987),327–339.[7]J.D.Axtell and K.H.Lee,Vertex Operator Algebras Associated to Type GAﬃne Lie Algebras,arXiv:1011.3473v1.[8] F.A.Bais and P.G.Bouwknegt,A classiﬁcation of subgroup truncations of thebosonic string,Nuclear Phys.B279(1987),561–570.[9]J.Bockenhauer,D.E.Evans and Y.Kawahigashi,Chiral structure of modularinvariants for subfactors,Comm.Math.Phys.210(2000),733–784.[10]R.E.Borcherds,Vertex algebras,Kac-Moody algebras,and the Monster,Proc.A83(1986),3068–3071.[11]Bourbaki,Groupes et alg`e bras de Lie,Hermann,Paris,1975.[12]P.Cellini,V.G.Kac,P.M¨o seneder Frajria and P.Papi,Decomposition rulesfor conformal pairs associated to symmetric spaces and abelian subalgebras of Z2-graded Lie algebras,Adv.Math.207(2006),156–204.[13] C.Dong,H.Li and G.Mason,Simple currents and extensions of vertex operatoralgebras,Comm.Math.Phys.180(1996)671–707.[14] C.Dong and G.Mason,On quantum Galois theory,Duke Math.J.86(1997),305–321.[15] D.E.Evans and T.Gannon,Modular invariants and twisted equivariant K-theory,Commun.Number Theory Phys.3(2009),209–296.[16] A.Felikson,A.Retakh and P.Tumarkin,Regular subalgebras of aﬃne Kac-Moody algebras,J.Phys.A41(2008),16pp.14DRAˇZEN ADAMOVI´C AND OZREN PERˇSE[17] E.Frenkel and D.Ben-Zvi,Vertex algebras and algebraic curves,MathematicalSurveys and Monographs,88,American Mathematical Society,Providence,RI, 2001.[18]I.Frenkel,Y.-Z.Huang and J.Lepowsky,On axiomatic approaches to vertexoperator algebras and modules,Mem.Amer.Math.Soc.104,1993.[19]I.Frenkel,J.Lepowsky and A.Meurman,Vertex Operator Algebras and the Mon-ster,Pure and Appl.Math.,Vol.134,Academic Press,Boston,1988.[20]I.Frenkel and Y.-C.Zhu,Vertex operator algebras associated to representationsof aﬃne and Virasoro algebras,Duke Math.J.66(1992),123–168.[21]J.Frohlich,J.Fuchs,I.Runkel and C.Schweigert,Algebras in tensor categoriesand coset conformalﬁeld theories,Fortsch.Phys.52(2004),672–677.[22]J.Fuchs,Aﬃne Lie algebras and quantum groups.An introduction with applica-tions in conformalﬁeld theory,Cambridge Monographs on Mathematical Physics.Cambridge:Cambridge University Press.xiv,433p.(1992).[23]J.E.Humphreys,Introduction to Lie algebras and representation theory,Grad-uate Texts in Mathematics,Vol.9.Springer-Verlag,New York-Berlin,1972.xii+169pp.[24]V.G.Kac,Inﬁnite dimensional Lie algebras,3rd ed.,Cambridge Univ.Press,Cambridge,1990.[25]V.G.Kac,Vertex Algebras for Beginners,University Lecture Series,SecondEdition,AMS,Vol.10(1998).[26]V.Kac and M.Wakimoto,Modular invariant representations of inﬁnite dimen-sional Lie algebras and superalgebras,A85(1988), 4956–4960.[27]V.Kac and M.Wakimoto,Classiﬁcation of modular invariant representations ofaﬃne algebras,in Inﬁnite Dimensional Lie algebras and groups,Advanced Series in Math.Phys.7,World Scientiﬁc,Teaneck NJ,1989.[28] A.Kirillov and V.Ostrik,On a q-analogue of the McKay correspondence andthe ADE classiﬁcation of sl2conformalﬁeld theories,Adv.in Math.171(2002), 183–227.[29]J.Lepowsky and H.Li,Introduction to vertex operator algebras and their repre-sentations,Progress in Math.,Vol.227,Birkhauser,Boston,2004.[30]H.-S.Li,Local systems of vertex operators,vertex superalgebras and modules,J.Pure Appl.Algebra109(1996),143–195.[31]H.Li,Extension of vertex operator algebras by a self-dual simple module,J.Algebra187(1997),236–267.[32]O.Perˇs e,Vertex operator algebras associated to type B aﬃne Lie algebras onadmissible half-integer levels,J.Algebra307(2007),215–248.[33]O.Perˇs e,Vertex operator algebra analogue of embedding of B4into F4,J.PureAppl.Algebra211(2007),702–720.[34]O.Perˇs e,Vertex operator algebras associated to certain admissible modules foraﬃne Lie algebras of type A,Glas.Mat.Ser.III43(63)(2008),41–57. [35]O.Perˇs e,Embeddings of vertex operator algebras associated to orthogonal aﬃneLie algebras,Comm.Algebra38(2010),1761–1778.[36] A.N.Schellekens and N.P.Warner,Conformal subalgebras of Kac-Moody alge-bras,Phys.Rev.D34(1986),3092–3096.[37]M.Wakimoto,Lectures on inﬁnite-dimensional Lie algebra,World Scientiﬁc,River Edge NJ,2001.F aculty of Science-Department of Mathematics,University of Zagreb, CroatiaE-mail address:adamovic@math.hr;perse@math.hr。

Generalized Polar Coordinates on Lie groups and Numerical

REPORTSININFORMATICSISSN0333-3590Generalized Polar Coordinates on Lie groups andNumerical IntegratorsS.Krogstad,H.Z.Munthe-Kaas,A.Zanna REPORT NO244April2003Department of InformaticsUNIVERSITY OF BERGENBergen,NorwayThis report has URL http://www.ii.uib.no/publikasjoner/texrap/pdf/2003-244.pdf Reports in Informatics from Department of Informatics,University of Bergen,Norway,is available at http://www.ii.uib.no/publikasjoner/texrap/.Requests for paper copies of this report can be sent to:Department of Informatics,University of Bergen,Høyteknologisenteret,P.O.Box7800,N-5020Bergen,NorwayGeneralized Polar Coordinates on Lie groups and NumericalIntegratorsS.Krogstad,H.Z.Munthe-Kaas,A.ZannaAbstractMotivated by recent developments in numerical Lie group integrators,we introduce a family of local coordinates on Lie groups denoted generalized polar coordinates .Fast algorithms are derived for the computation of the coordinate maps,their tangent maps and the inverse tangent maps.In particular we discuss algorithms for all the classical matrix Lie groups and optimal complexity integrators for n -spheres.1IntroductionLie group methods for integration of differential equations has been an active area of research over the last decade [7,20,24,17,22].Consider the family of integrators based on local coordinates as presented in [12](see Algorithm 1).These methods are expressed in terms of a local coordinate map Φfrom a Lie algebra to a Lie group and the inverse tangent map of Φ.Analytic coordinate maps include the exponential map [22],the Cayley map,and more generally diagonal Pad´e approximants to the exponential.It is well known that for certain groups (e.g.SL (n )),the only analytic map from the algebra to the group is the exponential mapping [18].Matrix splitting techniques yield non-analytic coordinate maps.Among these coordinates of the second kind ,studied by Owren and Marthinsen [25],show excellent computational cost for certain groups.Unfortunately,it is not known how this approach can be applied to e.g.the real orthogonal group.In this paper an other family of coordinates based on matrix splittings is studied.By recursively applying generalized polar decompositions of the Lie algebra [27,23],we obtain coordinates on all the classical matrix groups,where both the coordinate maps and the (forward and inverse)tangent maps can be computed efﬁciently.Let us brieﬂy review the generalized polar coordinates as deﬁned in Section 3.Consider a nested sequence of Lie algebrasg =g 0⊃g 1⊃···⊃g kderived from a sequence of linear maps σi :g i →g i ,i =0,...,k −1,such that the following relations holdσ2i =Iσi ([X,Y ])=[σi (X ),σi (Y )]Range 1(I +σi ) =g i +1.An arbitrary element Z ∈g is decomposed asZ =P 0+P 1+···+P k −1+K k ,(1.1)where the components are computed as follows.Let K 0=Z and for i =0,...,k −1letK i +1=12(I +σi )K i ∈g i +1P i =K i −K i +1=12(I −σi )K i .The coordinate maps to be studied in this paper are of the formΦ(Z )=exp(P 0)·exp(P 1)···exp(P k −1)·exp(K k ).(1.2)It is assumed that the computations in the last Lie algebra g k can be done fast,either because of low dimensionality or because it has a special structure (diagonal or block diagonal matrices).The computations involving the matrices P i and the tangent maps are relying on the structure of 2-cyclic matrices to be discussed in the sequel.The paper is structured as follows.In Section 2we review basic results of linear algebra,matrix Lie theory and numerical Lie group integrators.Section 3introduces the generalized polar coordinates,and develops the theory of their tangent maps.In Section 4the classical matrix groups,symmetric spaces and spheres are discussed in detail,and in Section 5we present numerical examples.2Preliminaries 2.1Coordinates in Lie group integratorsIn order to motivate the theory in the sequel,we will brieﬂy review a class of numerical integrators intro-duced in [21,22].We will present the theory in the concrete context of matrix Lie groups.The generaliza-tion to general Lie groups is discussed in [16].Let g denote a matrix Lie algebra ,i.e.a family of square matrices closed under linear combinations and matrix commutators,[A,B ]=AB −BA.Let G denote the Lie group of g ,deﬁned as the set of matrices obtained by taking matrix exponentials of g and products of these exponentials.The Lie group is closed under matrix products and matrix inversions.Let M denote a linear space and ·:G ×M →M an action of G on M ,deﬁned as a map satisfyingg ·(h ·y )=(gh )·y,for all g,h ∈G ,y ∈M .The action induces a product ·:g ×M →T M ,where T M denotes the tangents of M ,viaV ·y =d d t t =0exp(tV )·y.We consider differential equations evolving on M ,written in the form [22]y (t )=f (y (t ))·y,(2.1)where y (t )∈M and f :M →g .Let Φ:g →G denote a smooth mapping from a Lie algebra into a Lie group such that Φ(0)=e ,where e is the identity in G .Let d ΦZ denote the right trivialized tangent of Φat a point Z ∈g ,i.e.∂∂s s =0Φ(Z +sδZ )=d ΦZ (δZ )Φ(Z ).It follows easily that d ΦZ is a linear map from g to itself.We assume that d Φ0=I ,thus d ΦZ is invertible for Z sufﬁciently close to 0.The map Φdeﬁnes a diffeomorphism between a neighbourhood of 0∈g and a neighbourhood of e ∈G .Via the translations on G ,we may extend Φto an atlas of coordinate charts covering the whole of G .Using the action of of G on M ,we may also via Φobtain a coordinate atlas on M .For numerical algorithms based on such coordinates,it is often essential that we can compute both Φand the inverse tangent map d Φ−1Z efﬁciently.As an example,consider the class of numerical Lie group integrators for (2.1),introduced and developed in [22,12].Given y n ∈M and a timestep h ∈R .Given {a i,j }s i,j =1and {b j }s j =1,the coefﬁcients of an s -stage Runge-Kutta method.We step from y n ≈y (t n )to y n +1≈y (t n +h )as:Algorithm 1for i =1,s ,U i = s j =1a i,j ˜Kj K i =h ·f (Φ(U i )·y n )˜K i =d Φ−1U i (K i )end y n +1=Φ( s j =1b j ˜Kj )·y n Various coordinate maps Φhave been proposed and studied in the literature.The exponential mapping is used in [22]and the Cayley map in [20].Owren and Marthinsen [25]develop the theory of coordinates of the second kind .Given a basis {V i }for a d -dimensional Lie algebra g ,an element Z = d i =1z i V i maps toΦ(Z )=exp(z 1V 1)·exp(z 2V 2)···exp(z d V d ).Owren and Marthinsen introduce special classes of ‘nice’bases,so-called Admissible Ordered Bases (AOBs),and show that for such bases the map Φ(Z )and d ΦZ −1can be computed in O (d 3/2)opera-tions,d =dim g .Using the representation theory of semisimple Lie algebras,they show that for certain semisimple Lie algebras,AOBs can be obtained from Chevalley bases.In the cases where AOBs are found,they report favourable numerical experiments indicating that the resulting numerical algorithms are between two and six times faster than corresponding algorithms based on using Φ(Z )=exp(Z ).Unfortunately,it is not known if AOBs can be found for all classical matrix Lie groups.In particular there are still open problems with several of the real matrix groups,such as the real orthogonal group SO (n,R )with algebra so (n,R ),consisting of real skew symmetric matrices.In this note we will introduce coordinates based on generalized polar decompositions of g ,and we will show that this leads to fast computable coordinates for many Lie algebras,among these all the classical matrix Lie algebras.Furthermore,the theory of generalized polar coordinates is considerably simpler than the theory of second kind coordinates.The basis for our approach is some results from the theory of symmetric spaces,as given in [23].We will now review some linear algebra and basic elements of the theory of symmetric spaces needed for the present purpose.2.2Projections,involutions and 2-cyclic matricesBy a projection matrix on a vector space V we mean a linear map Π:V →V such that Π2=Π.Unless explicitly stated we will not require projections to be orthogonal (i.e.ΠT =Π).By an involutive matrix wemean a linear map S:V→V such that S2=I.These two concepts are naturally linked by the following lemma,whose proof is trivial.Lemma2.1To any projection matrixΠthere corresponds an involutive matrix S=I−2Π.Conversely,to any involutive S there corresponds two projection matricesΠ−S andΠ+Sdeﬁned byΠ−S =12(I−S)(2.2)Π+S =12(I+S).(2.3)These satisfy the following relationsΠ−S +Π+S=I(2.4)Π−S Π+S=Π+SΠ−S=0(2.5)SΠ−S =−Π−S(2.6)SΠ+S =Π+S.(2.7)Thus V splits in the direct sum of two subspaces,the±1eigenspaces of S,whereΠ±S are projectionsonto these.Note that if S is involutive then also−S is involutive,the latter corresponding to the opposite identiﬁcation of the+1and−1eigenspaces.Thus,at the moment there seems to be no fundamental difference between these two subspaces.We will,however,later return to involutive automorphisms on Lie algebras where the+1and−1eigenspaces play fundamentally different roles.Deﬁnition2.1A matrix K:V→V is block diagonal with respect to an involution S on V ifSKS=K,(2.8) and a matrix P:V→V is2-cyclic with respect to S ifSP S=−P.(2.9)Any matrix M can be split in a sum of a2-cyclic P and a block diagonal K,M=P+K,(2.10) whereP=12(M−SMS)(2.11)K=12(M+SMS).(2.12)To understand the structure of these matrices,it is convenient to represent linear operators M:V→V in 2×2block partitioned form in the following manner.The matrix M splits naturally in4parts:M=Π−S+Π+SMΠ−S+Π+S=Π−SMΠ−S+Π−SMΠ+S+Π+SMΠ−S+Π+SMΠ+S.(2.13)The partitioning of M with respect to S is deﬁned by dividing V in an upper block corresponding to therange ofΠ−S and a lower block corresponding to the range ofΠ+S.ThusM=M−−M−+M+−M++,(2.14)where M ij=Πi S MΠjSrestricted to the appropriate subspaces.In partitioned form S becomesS=−I00I.Thus,K consists of the diagonal blocks of M,while P is the off diagonal blocks.Efﬁcient computation of analytic functions of2-cyclic matrices will be crucial to our algorithms.Theorem2.2Let SP S=−P,where S2=I.LetΘ=P2Π−S.For an analytic functionψ(x)we have ψ(P)=ψ(0)I+ψ1(Θ)P+Pψ1(Θ)+Pψ2(Θ)P+ψ2(Θ)Θ,(2.15) whereψ1(x)=12√xψ(√x)−ψ(−√x)(2.16)ψ2(x)=12xψ(√x)+ψ(−√x)−2ψ(0)(2.17)and where we deﬁneΘ0≡Π−S.Proof.Involution of S implies SP=−P S henceΠ±S P=PΠ∓S.By induction it is now easy to verify thatfor any k≥0we haveP2k+1=Θk P+PΘk,P2k+2=Θk+1+PΘk P.Lettingψ(x)= ∞i=0αi x i,weﬁnd from these formulae thatψ(P)=α0I+∞k=0α2k+1Θk P+P∞k=0α2k+1Θk+∞k=0α2k+2Θk+1+P∞k=0α2k+2Θk P.We deﬁneψ1(x)= ∞k=0α2k+1x k andψ2(x)=∞k=0α2k+2x k and derive(2.16)and(2.17)by straight-forward manipulation of the series forψ(x).2 Note that if P is partitioned with respect to S asP=0BA0,(2.18)thenΘ=P2Π−S =BA000.(2.19)Thusψi(Θ)=ψi(BA)00ψi(0)I.Since BA is a p×p matrix,p=Rank(Π−S ),we are mainly interested in cases where p is small,and thecase p=1is particularly important.Corollary2.2.1If p=1thenΘ=θΠ−S,for a scalarθ,andψ(P)=ψ(0)I+ψ1(θ)P+ψ2(θ)P2,(2.20) whereψ1andψ2are given in(2.16)-(2.17).Proof.IfΘ=θΠ−Sthenψ1(Θ)P+Pψ1(Θ)=ψ1(θ)Π−SP+PΠ−S=ψ1(θ)P.SimilarlyPψ2(Θ)P+ψ2(Θ)Θ=ψ2(θ)PΠ−SP+Π−SP2=ψ2(θ)P2.22.3Computation of analytic matrix functionsLet A:V→V be a matrix with a(small)number of different eigenvaluesλ1,...,λd.Letψ(x)be an analytic function.For later application,we will discuss some algorithms for computingw=ψ(A)v(2.21) for an arbitrary vector v∈V.2.3.1Via eigenspace projectionsIn the following we assume that A is non-defect(all Jordan blocks of size1).For i=1,...,d,letΠi denote projection onto the eigenspace corresponding toλi.Thendi=1Πi=IAΠi=Πi A=λiΠi.Thus(2.21)can be computed asw=ψ(A)v=di=1ψ(λi)Πi v.(2.22)The eigenspace projections can be expressed in terms of the left and right eigenvectors.Let x i,k and y i,k denote all the right and left eigenvectors of A,where i=1,...,d and k=1,...,Rank(Πi).These can be chosen such thatA x i,k=λi x i,k(2.23)y T i,k A=λi y T i,k(2.24)y T i,k x j,l=δi,jδk,l.(2.25)In other words,if x i,k are the columns of the eigenvector matrix X,then y Tj,l are the rows of X−1.WehaveΠi=kx i,k y T i,k.(2.26) We will return to other ways of representing these projections in the sequel.2.3.2Via the minimal polynomialAssuming A is non-defect,the d-degree polynomialq(x)=Πd j=1(x−λj)is the minimal polynomial of A,i.e.the lowest degree monic polynomial for which q(A)=0.Let r(x)=ψ(x)mod q(x)be the division remainder,deﬁned as the degree d−1polynomial such thatψ(x)=q(x)s(x)+r(x),where s(x)does not have singularities inλj.Ifψ(x)is a polynomial,the remainder r(x)can be computed by polynomial division.In the general analytic case,r(x)can be found from polynomial interpolation in the d pointsr(λi)=ψ(λi),for i=1,...,d.Since q(A)=0we see that w=ψ(A)v=r(A)v can be computed by the work of d−1matrix-vector products A v.In the general case,where A might be defect,let m j denote the size of the largest Jordan block associated with the eigenvalueλj.The minimal polynomial is now given asq(x)=Πd j=1(x−λj)m j.We obtain r(x)by Hermite interpolation.Inλj we let r(x)interpolateψ(x)and its derivatives up to order m j−1.The Newton form of r(x)is convenient to work with.Let divided differences be deﬁned asψ[λi]=ψ(λi)(2.27)ψ[λi,...,λi+k]=ψ[λi+1,...,λi+k]−ψ[λi,...,λi+k−1]λi+k−λifor distinctλj(2.28)ψ[λj,λj,...,λj]=1(m j−1)!ψ(m j−1)(λj)forλj repeated m j times.(2.29)Then r(x)is given asr(x)=ψ[x1]+(x−x1)ψ[x1,x2]+···+(x−x1)(x−x2)···(x−x m−1)ψ[x1,x2,...,x m],(2.30)where m= di=1m i and{x1,x2,...,x m}={λ1,...,λ1,λ2,...,λ2,...,λd,...,λd}(λi repeated m i times).Using this form,we obtain the following simple algorithm to compute z=ψ(A)v=r(A)v using m−1 matrix vector products.Algorithm2(Computing w=ψ(A)v)w:=ψ[x1]vfor i=1,...,m−1v:=A v−x i vw:=w+ψ[x1,...,x i+1]vend2.3.3Via Schur decompositionsIf A is far away from a normal matrix,or if it is defect,the eigenspace projection approach is numerically unstable.A better class of algorithms is based on transforming A to an upper triangular matrix by the Schur decomposition.Functions of upper triangular matrices can be computed by the algorithm of Parlett[13],anda general purpose algorithm including these steps is given in[8].We return to this approach in Section3.3.2.4Involutive automorphisms on Lie algebrasAn involutive automorphism on a Lie algebra g is an involutive mapσ:g→g such thatσ([U,V])=[σ(U),σ(V)].(2.31) Corresponding to the automorphismσon g there is an automorphism on the Lie group G,which plays an important role in the theory of symmetric spaces.In this paper we will,however,not need this automor-phism on G and we omit this from the discussion.Note that ifσis an automorphism,then−σis not an automorphism.Thus in this case we can clearly distinguish between the+1and−1eigenspaces ofσ,they play a fundamentally different role in the theory. LetΠ±σbe projections on±eigenspaces ofσ,given in(2.2)-(2.3).Denote these spaces byp=Range(12(I−σ))={P∈g|σ(P)=−P}k=Range(12(I+σ))={K∈g|σ(K)=K}.The subspace k is a Lie subalgebra of g,while p forms a Lie triple system,meaning that it is closed under double brackets[P1,[P2,P3]]∈p for all P1,P2,P3∈p.More generally,the spaces p and k satisfy the following odd-even parity rules under brackets(compare to multiplication table of-1and1):[k,k]⊆k(2.32)[p,k]⊆p(2.33)[p,p]⊆k.(2.34) The decomposition we have introduced is thus closely related to the so called Cartan decomposition, see[14].However,the Cartan decomposition requiresσto be a Cartan involution,whose deﬁnition involves a certain non-degeneracy of a bilinear form derived from the Cartan-Killing form.For the applications in this paper the Cartan property ofσis not needed,thus we have considerable freedom in choosing a suitable σ.Corresponding to the additive splitting g=p⊕k there exists a multiplicative splitting of G.Any element g∈G sufﬁciently close to I can be written as a productg=exp(P)exp(K),where P∈p and K∈k.The element exp(P)∈G belongs to a so called symmetric space,while exp(K)belongs to a Lie subgroup of G.For example,if G=GL(n)andσ(Z)=−Z T,then the splitting g=p+k corresponds to writing a general matrix as the sum of a symmetric and a skew matrix.The corresponding product splitting of G is the polar decomposition,where a matrix is written as a product of a symmetric positive deﬁnite matrix and an orthogonal matrix.In general,ifσis any involutive automorphism,we will refer to the decomposition as a generalized polar decomposition,see[23]for more details.3Generalized Polar Coordinates(GPC)The coordinates we are studying in this paper are recursively deﬁned as follows.Deﬁnition3.2GPC denotes any coordinate map from a Lie algebra to a Lie group obtained by the follow-ing recursion:•The exponential map exp:g→G is GPC on G.•Letσbe an involutive automorphism on g and letΠ±σbe deﬁned by(2.2)-(2.3).Let k=Range(Π+σ)⊂g and Gσ⊂G be the corresponding sub-algebra and subgroup.If˜Φ:k→Gσis GPC on Gσthen amapΦ:g→G deﬁned asΦ(Z)=expΠ−σZ·˜ΦΠ+σZ,(3.1)is GPC on G.The coordinates are completely determined by a sequence of involutive automorphisms{σi}k−1i=0,givingrise to a sequence of subalgebras g=g0⊃g1⊃···⊃g k,where g i+1=Range(Π+σi )andσi:g i→g i.By unfolding the recursion,we obtain the equivalent form of the coordinate map given in(1.1)-(1.2).Note that we let the-1eigenspace appear on the left and the+1on the right in(3.1).This is important if we want to compute right trivialized tangents.If we instead want to work with left trivializations,we must reverse the deﬁnition and let the+1eigenspace appear on the left.As a trivial example consider G=C∗(the multiplicative group of nonzero complex numbers),letσ(z)=¯z (complex conjugation)and let˜Φ(K)=exp(K).If Z=X+iY thenΠ−σZ=X,Π+σZ=iY and Φ(Z)=exp(X)exp(iY)=r exp(iθ),where r=exp(X)andθ=Y.This yields(classical)polar coordinates on C∗.3.1The coordinate mapTo obtain efﬁcient algorithms for computing the coordinate map,we assume thatσis a low rank inner automorphism,deﬁned as follows.Deﬁnition3.3An automorphismσ:g→g of the formσ(Z)=SZS,where S2=I(3.2) is called an inner automorphism.1By the rank of an inner automorphism we mean p=Rank(Π−S).Ifσis given by(3.2)then P=Π−σ(Z)=12(Z−SZS)is2-cyclic with respect to S,i.e.SP S=−P.Theorem2.2yields the following formula for computing exp(P).Theorem3.3Let SP S=−P,where S2=I.LetΘ=P2Π−S,thenexp(P)=I+ψ1(Θ)P+Pψ1(Θ)+Pψ2(Θ)P+ψ2(Θ)Θ,(3.3) 1If G is a matrix group,thenσinduces the group automorphism G g→SgS∈G.Although it is not necessary that S∈G,it must be an element of some larger group containing G as a subgroup,and on this larger group the automorphism is properly of inner type.We stick to the name‘inner’also when S∈G.whereψ1(x )=sinh(√x )√x=sin(√−x )√−xfor x =01for x =0(3.4)ψ2(x )= 2sinh 2(√x/2)x =2sin 2(√−x/2)−x for x =012for x =0(3.5)Proof.From (2.16)we ﬁnd ψ1(x )=(exp(√x )−exp(−√x ))/(2√x )=sinh(√x )/√x .Similarly from (2.17)ψ2(x )=(cosh(√x )−1)/x ,the equivalent form (3.5)being numerically better for small x .2See notes after Theorem 2.2for a discussion of the structure of Θ.Combined with the algorithms in Section 2.3for computing ψi (Θ),we have practical algorithms to be investigated in the sequel.For the rank 1case we establish explicit formulae.Corollary 3.3.1Let SP S =−P ,where p =Rank(Π−S )=1.Let the scalar θbe given as Π−S P 2=θΠ−S .Then exp(P )= I +P +12P 2if θ=0I +sin(√−θ)√−θP +2sin 2(√−θ/2)−θP 2if R θ<0I +sinh(√θ)√θP +2sinh 2(√θ/2)θP 2if R θ>0or if θis complex.(3.6)It should be noted that this is exactly the same formula as the Rodrigues formula for skew 3×3matrices.Indeed,let P = x= 0−x 3x 2x 30−x 1−x 2x 10and let S =I −2zz T /z T z ,where z is a vector such that z T x =0.Then SP S =−P .One ﬁnds thatθ=−x T x ,from which Rodrigues formula follows.3.2The tangent mapIn this section we will develop the formulae for the computing tangent map of Φand its inverse.Whereasthe theory of the previous section regarded elements of g as matrices (linear operators in R n ×nacting on R n ),we are now concerned with d ΦZ and d Φ−1Z which belong to the space of all linear operators from g to itself,denoted End (g ).If g is represented as n ×n matrices,then End (g )could be represented as n 2×n 2matrices.Inversion of such linear operators by Gaussian elimination would cost O (n 6)operations,but we are seeking algorithms of complexity at most O (n 3).To achieve this,we can not rely on a matrix representation of End (g ),but rather work directly with operators (projections,involutions)as outlined in Section 2.2.The theory of this section relies on the odd-even splitting of the subspaces g =p ⊕k in (2.32)-(2.34).We will develop several formulae for the tangent map of Φand its inverse.In the following,let σbe an involutive automorphism on g ,and let P ∈p ⊂g ,i.e.σ(P )=−P .Deﬁne the linear operator ad P :g →g asad P (Z )=[P,Z ].(3.7)Theorem 3.4Let Z =P +K =Π−σZ +Π+σZ .The right trivialized tangent of Φand its inverse are given asd ΦZ=d exp p Π−σ+Ad exp(P )d ˜ΦK Π+σ= exp(u )−1uΠ−σ+exp(u )Π+σ Π−σ+d ˜ΦK Π+σ(3.8)d ΦZ −1= Π−σ+d ˜ΦK −1Π+σ 1+(u −1)cosh(u )sinh(u )Π−σ+(1−u )Π+σ,(3.9)where u =ad P .Proof.The ﬁrst form of (3.8)follows fromd ΦZ (δZ )Φ(Z )=∂∂s s =0exp(P +sδP )˜Φ(K +sδK )=d exp P (δP )exp(P )˜Φ(K )+exp(P )d ˜ΦK (δK )˜Φ(K )= d exp P (δP )+exp(P )d ˜ΦK (δK )exp(−P )Φ(Z ).Let u =ad P .Using d exp P =(exp(u )−1)/u ,Ad exp(P )=exp(ad P ),Π−σ2=Π−σand Π+σΠ−σ=0,we obtain the second form of (3.8).To verify (3.9),we observe from (2.32)-(2.34)that if ψ(x )is an analytic function with odd and even parts ψ(x )=ψo (x )+ψe (x ),ψo (x )=12(ψ(x )−ψ(−x )),ψe (x )=12(ψ(x )+ψ(−x )),thenΠ−σψ(u )Π−σ=ψe (u )Π−σ,Π+σψ(u )Π−σ=ψo (u )Π−σΠ−σψ(u )Π+σ=ψo (u )Π+σ,Π+σψ(u )Π+σ=ψe (u )Π+σ.ThusΠ−σexp(u )−1uΠ−σ=sinh(u )uΠ−σΠ+σexp(u )−1uΠ−σ=cosh(u )−1uΠ−σΠ−σexp(u )Π+σ=sinh(u )Π+σΠ+σexp(u )Π+σ=cosh(u )Π+σ.Using these formulae,we ﬁnd that1+(u −1)cosh(u )sinh u Π−σ+(1−u )Π+σ exp(u )−1uΠ−σ+exp(u )Π+σ =Π−σ+Π+σ=I.Since ˜Φis acting only on the subalgebra k we have Π−σd ˜Φk Π+σ=0and Π+σd ˜Φk Π+σ=d ˜ΦK Π+σfrom whichwe get Π−σ+d ˜ΦK −1Π+σ Π−σ+d ˜ΦK Π+σ =Π−σ+Π+σ=I,thus (3.9)is veriﬁed.2Using the same odd-even parity argument as in the proof of Theorem 3.4we obtain the following result:Corollary3.4.1The partitioning of dΦZ and dΦZ−1with respect toσisdΦZ=sinh(u)usinh(u)cosh(u)−1ucosh(u)I00d˜ΦK(3.10)dΦZ−1=I00d˜Φ−1Ku cosh(u)sinh(u)−u1−cosh(u)sinh(u)1,(3.11)where u=ad P and P=Π−σZ restricted to the appropriate subspace.For efﬁcient computation of the tangent map,it is essential to develop fast algorithms for computing analytic functions of ad P.First we use the theory of2-cyclic matrices to simplify(3.9).Lemma3.5Ifσ(P)=−P,then ad P is2-cyclic with respect toσ,i.e.σad Pσ=−ad P.(3.12)Proof.Let Z∈g be arbitrary.We haveσad Pσ(Z)=σ([P,σ(Z)])=σ(P),σ2(Z)=[−P,Z]=−ad P(Z).2Theorem3.6Letσbe an arbitrary involutive automorphism.Let Z=P+K where P=Π−σZ,let u=ad p andΩ=ad2PΠ−σ.ThendΦZ−1=Π−σ+d˜ΦK−1Π+σI+uψ1(Ω)Π−σ−Π+σ+ψ2(Ω)Π−σ,(3.13)whereψ1(x)=−tanh(√x/2)√x=−tan(√−x/2)√−xfor x=0−12for x=0(3.14)ψ2(x)= √−xtanh(√−x)−1=√xtan(√x)−1for x=00for x=0.(3.15)Proof.Starting from(3.9),we use Lemma3.5with Theorem2.2.Since uΠ−σ=Π+σu,andΠ+σΠ−σ=0,all terms of the formψi(Ω)uΠ−σvanish,and the result follows by a straightforward symbolic computation.2To employ the algorithms of Section2.3,we must understand the eigenspace structure ofΩ.To use Algo-rithm2,we use the following result which is an immediate consequence of Theorem3.9below.Note that µi are the eigenvalues of the p×p matrix BA in(2.19).Corollary3.6.1Letσ(P)=SP S=−P,where Rank(Π−S )=p.LetΘ=P2Π−Sbe nondefect of rank p,with d≤p different non-zero eigenvalues{µi}d i=1.ThenΩ=ad2pΠ−σis non-defect with1+d+d2different eigenvalues given as0,{µi}i ,√µi−√µji>j,√µi+√µji≥j.(3.16)Thus,the minimal polynomial ofΩis the monic1+d+d2-degree polynomial with zeros in the points (3.16).In the rank-1case,we can obtain explicit formulae.If the scalar θgiven by P 2Π−S =θΠ−S is nonzero,thenthe minimal polynomial for Ω=ad 2p Π−σbecomesq (x )=x (x −θ)(x −4θ).From (3.13)we employ Algorithm 2symbolically to obtain:Corollary 3.6.2Let σbe a rank-1involutive automorphism.Let Z =P +K where P =Π−σZ ,let u =ad pand let θbe the scalar P 2Π−S =θΠ−S ,as in Corollary 3.3.1.If θ=0thend ΦZ −1= Π−σ+d ˜ΦK −1Π+σ I +u (ψ1(θ)+ψ2(θ)u 2)Π−σ−Π+σ − ψ3(θ)u 2+ψ4(θ)u4 Π−σ (3.17)whereψ1(θ)=−8tanh(√θ/2)+tanh(√θ)6√θψ2(θ)=2tanh(√θ/2)−tanh(√θ)6θ32ψ3(θ)=15+√θ(tanh(√θ)−15coth(√θ))12θψ4(θ)=−3+√θ(−tanh(√θ)+3coth(√θ))12θ2.It should be noted that u 2and u 4act on p =Range (Π−σ),yielding also a result in p .If g ⊂Rn ×nthen the cost of computing the terms u 2and u 4is O (n ).The main work is the computation of the single u acting on p ⊕k ,with a cost of O (n 2).Now we return to the general rank p case and the computation of ψ(Ω)via eigenspace projections.Let P and Ωbe as in Lemma 3.6.1.For our present purpose it is not important whether or not µi are distinct,so we just assume that Θis non-defect with p eigenvalues µi for i =1,...,p .Let the corresponding left and right eigenvectors be denoted y T i and x i ,normalized such thaty T i Ω=µi y T iΩx i =µi x i y T i x j=δi,j .The following is veriﬁed by straightforward computation.Lemma 3.7P has 2p nonzero left and right eigenvectors given asv ±i =1√2(x i ±P x i /√µi )for i =1,...,p (3.18)w T ±i=1√2y T i ±y T i P/√µi for i =1,...,p ,(3.19)satisfyingP v ±i=±√µi v ±iw T ±i P =±√µi w T ±i w T j v k=δj,kfor j,k ∈{±1,...,±p }.Lemma 3.8If Θ=P 2Π−S non-defect and of rank p,then P is non-defect.Proof.In Lemma 3.7we found 2p independent right eigenvectors v ±i .Referring to the partitioned form of P in (2.18),we have that B is a p ×(n −p )matrix with linearly independent rows.Thus there are n −2pzero right eigenvectors of P of the form 0x T T,where x ∈Ker (B ).We have found n independent right eigenvectors of P .2For i =1,...,d let the eigenspace projections of P be given asΠ±i =v ±i w T ±i (3.20)Π0=I −d i =1(Πi +Π−i ).(3.21)Now we are ready to formulate the main theorem describing the eigenspace structure of Ω.Note that k ⊂ker (Ω)and Ω(p )⊂p ,thus we can restrict the discussion to the action of Ωon p .Theorem 3.9Let σbe a rank p inner automorphism,and let σ(P )=−P .If Θ=P 2Π−S is non-defect and of rank p ,then Ω=ad 2P Π−σis non-defect with a complete list of eigenspace projections Πi,j :p →p expressed in terms of an arbitrary W ∈p asΠ0,0(W )=Π0W Π0(3.22)Πi,0(W )=(Πi +Π−i )W Π0+Π0W (Πi +Π−i )for i =1,...,p (3.23)Πi,j (W )=Πi W Πj +Π−i W Π−j for i,j =1,...,p and i >j (3.24)Πi,−j (W )=Πi W Π−j +Π−i W Πj for i,j =1,...,p and i ≥j .(3.25)The corresponding eigenvalues are given viaΩΠ0,0(W )=0(3.26)ΩΠi,0(W )=µi Πi,0(W )(3.27)ΩΠi,j (W )=(√µi −√µj )2Πi,j (W )(3.28)ΩΠi,−j (W )=(√µi +√µj )2Πi,−j (W ).(3.29)Proof.For Πi,j we computead P (Πi W Πj )=P Πi W Πj −Πi W Πj P =(√µi −√µj )Πi W Πj .Hence under the action of ad 2P the two projections W →Πi W Πj and W →Π−i W Π−j correspond to the same eigenvalue√µi −√µj 2.These can be combined into the single projection Πi,j (W )=Πi W Πj +Π−i W Π−j .Now,since P Π±i =±√µi Π±i and σ(P )=−P ,we ﬁnd that σ(Π±i )=Π∓i .If W ∈p we have σ(W )=−W and henceσ(Πi,j (W ))=σ(Πi W Πj +Π−i W Π−j )=Π−i (−W )Π−j +Πi (−W )Πj =−Πi,j (W ).Thus W ∈p ⇒Πi,j (W )∈p .We conclude that Π−σΠi,j (W )=Πi,j (W ),which yieldsΩΠi,j (W )=(√µi −√µj )2Πi,j (W ).。

V-Algebras and Their Free Field Realizations

V-Algebras and Their Free Field RealizationsYi Cheng and Lin ZhangDepartment of MathematicsUniversity of Science and Technology of ChinaHefei,Anhui230026,P.R.ChinaE-mail:chengy@,lzhang@AbstractThe V-algebras are the non-local matrix generalization of the well-known W-algebras.Their classical realizations are given by the second Poisson brackets associated with the matrix pseudodiﬀerential operators.In this paper,by using the general Miura transformation,we give the decomposition theorems for the second Poisson brackets,from which we are able to construct the freeﬁeld realizations for a class of V-algebras including V(2k,2)-algebras that corresponds to the Lie algebra of C k-type as the particular examples. The reduction of our discussion to the scalar case provides the similar result for the W BKP-algebra.1Recently,Bilal [1-3]has proposed non-local matrix generalizations of W -algebras,called V (m,n )-algebras.Their classical realizations appears naturally as the second Hamiltonian structures associated with the matrix version of the m th -order diﬀerential operatorsL =∂m +m −2j =0U j ∂j (1)where U j are n ×n matrix valued functions.The simplest example is the V (2,2)-algebra.It arises in the study of the non-abelian Toda ﬁeld theory as a model for strings propagating on a black hole background [1]and its Poisson bracket version can be given by the second Hamiltonian structure associated with [2]L =∂2−U,U = T −√2V +−√2V −T(2)The free ﬁeld realization of the V (2,2)-algebra was constructed ﬁrst by the factorizationL =∂2−U =(∂+P )(∂−P )(3)such thatU =P +P 2(4)and then by expressing P in terms of the vertex-like operators.In analogy to the scalar case,(4)is called the Miura transformation.In general,the Miura transformation can be given by the similar factor-ization [3]L =∂m +m −2j =0U j ∂j =m j =1(∂+P j )(5)and relates the second Poisson bracket of U j to much simpler ones of P j ,where P j are n ×n matrices and satisfy the constraint P j =0.However to the contrary of the scalar (i.e.n =1)case,the P j are not free ﬁelds in general,and as far as we know,except V (2,2)-algebras,it is not clear how to give the free ﬁeld realization for general V (m,n )-algebra.As pointed by Bilal [3],the reason of this diﬀerence between the W -algebras and their matrix generalization V -algebras is the existence of non-local terms in the V -algebras.2In this paper,we ﬁrst generalize Bilal’s V (m,n )-algebra to the V -algebra associated with the m th -order matrix pseudodiﬀerential operator (matrix ΨDO)L =∂m +m −1j =−∞U j ∂j (6)where U j are n ×n matrix valued functions,then give a general decomposition theorem for the second Poisson bracket associated with (6)by using the factorization L =L 1L 2,where both L 1and L 2are matrix ΨDOs with the order being m 1and m 2respectively and satisfying m 1+m 2=m .It is not diﬃcult to generalize the factorization to the rational form L =L 1L −12since we may think that L 1is factorized by L 1=LL 2.The above discussion is nothing but matrix generalization of our previous work on the scalar case[4,5].Thirdly,we consider the free ﬁeld realization of a more general class of V -algebras that correspond to the L of (6)with 2×2matrix coeﬃcients and satisfying L =L ∗for a proper deﬁned adjoint action of the matrix ΨDOs.Finally,as a consequence,when we restrict to the scalar case,we obtain the W -algebras represented by the second Poisson brackets of the BKP hierarchy and their free ﬁeld realization.Let L in (6)be the m th -order matrix ΨDO.For any functionalf = f (U m −1,U m −2,···)dxwe deﬁne δf δL =m −1 j =−∞∂−j −1δf δU j(7)and (δf δU j )αβ=∞ r =0(−1)r ∂r ∂x r (∂f ∂(U (r )j )βα)(8)is the matrix version of the Euler variation,where (U (r )j )βαdenotes the (β,α)matrix element of r th derivative of U j .Using (7)we ﬁndd f =<δf δL,δL >(9)where the product <·,·>is deﬁned by <A,X >=tr res AXdx (10)3for any two matrix ΨDOs of the form A = m −1j =−∞A j ∂j and X =m −1j =−∞∂−j −1X j .As in the scalar case,A corresponds to the “vector ﬁeld”∂A and X is called “one form”paired by (10)with the vector ﬁeld.The residues in (10)is deﬁned to be the coeﬃcient of ∂−1term.According to [3],The second Poisson bracket associated with (6)can be deﬁned in analogy with the scalar case [6]{ f, g }L =<H (δf δL ),δg δL >(11)where H (X )=(LX )+L −L (XL )+=L (XL )−−(LX )−L (12)mapping an one form X = m −1j =−∞∂−j −1X j to the vector ﬁeld ∂H (X ),where the subscripts “±”are understood as the pure diﬀerential part or the residual part of the ΨDO.If U m −1=0is assumed the following conditionres[δf δL ,L ]=0(13)must be taken into account such that the leading coeﬃcient of δf is expressedin terms of others.The second Poisson bracket (11)constrained to U m −1=0is called the V -algebra [1-3].The bracket (11)is bilinear and anti-symmetric because of the apparent properties of the product (10).It will follow from the results on the Miura transformation that for a class of matrix ΨDOs used in this paper,(11)also obeys the Jacobi identity.Nevertheless the following theorem does not depend on the property of the Jacobi identity.Theorem 1By the factorizationL =L 1L 2(14)whereL i =∂m i +m i −1j =−∞U ij ∂j ,i =1,2(15)are m th i -order matrix ΨDOs with m 1+m 2=m ,then the Poisson bracket associated with L of (6)is decomposed to the summation of two brackets that are associated with L 1and L 2respectively{ f, g }L ={ f, g }L 1+{ f, g }L 2(16)4If U m−1=U m1−1+U m2−1=0is assumed,then(13)is equivalent to res[δfδL1,L1]+res[δfδL2,δL2]=0(17)The proof of this theorem is essentially the same as we shown for the scalar case in[4.,5],i.e.by(14)any functional f of U j is also a functional of U1j and U2j,therefore on the one hand we haved f=tr resδfδLδLdx=tr resδfδL(δL1L2+L1δL2)dxand on the other handd f=tr res(δfδL1δL1+δfδL2δL2)dxThe above two expression imply thatδf δL1=L2δfδL,δfδL2=δfδLL1(18)each of them modular an(−m1−1)th-order and(−m2−1)th-order matrix ΨDO respectively.Substitute(18)to the right hand side of(16)and by the same calculation as that in[4,5],we can prove the theorem.It is easy to generalize Theorem1to the factorization L=L1···L r,in particular if r=m and L j=∂+P j,we immediately recover the result ofBilal[3]{ f, g}L=mj=1tr res[δfδP j,∂+P j]δgδP jdx(19)since the second Poisson bracket associated with L j=∂+P j is simply{ f, g}Pj =tr res[δfδP j,∂+P j]δgδP jdx(20)The constraint U m−1=P j=0is then equivalent tomj=1(δfδP j) =0(21)5Theorem 2If L =L 1L −12(22)where for the simplicity we assume that L 1and L 2are (m +k )th -order and k th -order matrix diﬀerential operators respectively,then{ f, g }L ={ f, g }L 1−{ f, g }L 2(23)The proof of this theorem can be completed simply by considering that L 1=LL 2is factorized and then by applying Theorem 1.The scalar version of the factorization L =L 1L −12was appeared in [7,8]for the study of W -algebras.In the following we discuss the reduction of the second Poisson bracket(11)to the subspace of matrix ΨDOs that satisfy L =L ∗.For the matrix ΨDOs A = A j ∂j ,we deﬁne the matrix version of the adjoint action on A by A ∗= (−∂)j σA T j σ−1(24)where “T ”denotes the matrix transposition,σis an n ×n constant matrix such that the adjoint action satisﬁes(A ∗)∗=A(AB )∗=B ∗A ∗(A ∗)+=(A +)∗ tr res A ∗dx =− tr res Adx (25)It is easy to see that such a matrix can be chosen freely as long as σis symmetric.Let W =L −L ∗= W j ∂j (26)where W j =U j −m −1 i =j (−1)i i i −j σ∂i −j U T i ∂x i −j σ−1(27)then we can calculate thatδ(W j )αβδL =(E βα−(−1)j σE αβσ−1)∂−j −1(28)6are symmetricδ(W j )αβδL =(δ(W j )αβδL)∗(29)with respect to the matrix version of adjoint action,where E αβare the n ×n matrices only with the (α,β)th matrix element being equal to one and others to zero.If we suppose that m =2k and L is symmetricL =L ∗(30)(i.e.W j =0),then the “vector ﬁelds”∂A on the submanifold W j =0will be parametrized by the deformations of L that remain symmetric.These A are clearly the matrix ΨDOs of order at most 2k −1obeying the symmetric property A =A ∗.The “one forms”X = ∂−j −1X j on the submanifold W j =0must be chosen to be those which are mapped via the Hamiltonian map H to the vector ﬁelds ∂H (X )tangent to the submanifold W j =0.In other words,(H (X ))∗=H (X ).Since(H (X ))∗=−H (X ∗),(31)X must be anti-symmetric X =−X ∗modular a (−m −1)th order of matrix ΨDO (i.e.the kernel of H ).It can easily be checked that these one forms are nondegenerately paired with the vector ﬁelds ∂A ,A =A ∗.Actually we have checked that for some simple cases for any functional f= fdx restricted on W j =0,X =δf really satisﬁes X =−X ∗modular the kernel of H .Therefore the Poisson bracket of two functionals f= fdx and g = gdx on the submanifold can be given by{ f,g }L =1<H (δf −(δf )∗),δg −(δg )∗>=12<H (δf δL −(δf δL )∗),δg δL >(32)with L being symmetric.The above argument is an analogue of that for the supersymmetric BKP hierarchy [9].The following theorem will provide another argument.Theorem 3.If the m th -order (m =2k )symmetric matrix ΨDO L is factorized byL =L ∗1L 1(33)7withL 1=∂k +k −1 j =−∞V j ∂j (34)then we have { f, g }L =12{ f, g }L 1(35)Proof :Any functional of U m −1,U m −2,···is also a functional of V k −1,V k −2,···via the relation of (33).Therefored f =<δf δL ,δL >=<δf δL,δL ∗1L 1+L ∗1δL 1>(36)d f =<δf δL 1,δL 1>so δf δL 1=((δf δL )−(δf δL)∗)L ∗1(37)modular a (−k −1)th -order matrix ΨDO.Substitute this expression to the Poisson bracket {·,·}L 1with respect to L 1we have the Poisson bracket (32)with respect to L ,which can be expressed by (35).We may continue to factorize L 1L 1=l j =1(∂+P j )−1k +2l j =l +1(∂+P j )(38)where l is an arbitrary integer,and then apply Theorem 2and 3,we ﬁnd that the Poisson bracket (32)of L =L ∗1L 1becomes{ f, g }L =12k +2l j =l +1{ f, g }P j −12l j =1{ f, g }P j (39)with each { f, g }P j being given by (20).Let us now calculate the coeﬃcient of the second leading term of L =L ∗1L 1with L 1being in (38).It isU m −1=(−1)k k +2l j =l +1(P j −σP T j σ−1)−(−1)l l j =1(P j −σP T j σ−1)(40)8We immediately ﬁnd that a suﬃcient condition of U m −1=0isP j −σP T j σ−1=0,j =1,2,···,k +2l (41)namely the restriction of the Poisson bracket (32)of L to the submanifold U m −1=0can be realized if each copy of the Poisson bracket in the form of(20)associated with ∂+P j can be restricted to the submanifold of (41).According to the above analysis,we choose n =2,σ=0110 (42)and k +2l copies of Bilal’s V (2,2)-algebraP j = T j −√2V +j −√2V −j T j ,1≤j ≤k +2l (43)among them the ﬁrst l copies have a sign diﬀerence with the V (2,2)-algebra.It is obvious that P j obey (41)and their elements can be expressed in terms of k +2l independent groups of vertex-like ﬁelds.Thus we may construct the free ﬁeld realization of the V -algebra that corresponds to the second Poisson bracket on the space of m th -order (m =2k )and 2×2matrix ΨDOs restricted by L =L ∗.A simple case is for l =0,i.e.if L 1is a pure diﬀerential operator,soL =L ∗1L 1=(−1)k (∂−P k )···(∂−P 1)(∂+P 1)(∂+P k )(44)the V (2k,2)-algebra in this case corresponds to the Lie algebra of the C k -type[3].Our result give its free ﬁeld realization.Note that from mathematical point of view,the (−1)k factor does not aﬀect the structure of Poisson bracket essentially.Finally we are going to restrict the above results to the scalar case n =1and connect the Poisson bracket (32)for n =1with the Poisson bracket for the BKP hierarchy.LetΛ=∂2k +1+2kj =−∞v j ∂j (45)where v j are scalar functions.Then we deﬁneL =∂Λ=∂2k +2+2k +1j =−∞u j ∂j (46)9The relation between v j and u j can be given explicitlyu 2k +1=v 2k u j =v j +v j −1(47)from which we ﬁrst have ∂f∂v (l )j=∂f ∂u (l )j +1+∂f ∂u (l −1)j and so δf δv j =δf δu j +1−(δf δu j) which implies that δf δL =δf δΛ∂−1+(a (−2k −3)th -order ΨDO )(48)If we assume that Λis the ΨDO associated with the BKP hierarchy [10]Λ∗=−∂Λ∂−1(49)then L in (46)is symmetric L =L ∗and u 2k +1=v 2k =0where the adjoint action on the scalar ΨDO A = a j ∂j is deﬁned as usual A ∗= (−∂)j a j .Substitute (48)into (32)for n =1we notice that the second term of the right hand side does mot contribute anything and the Poisson bracket in terms of Λis given by{ f, g }Λ= res[∂−1(∂Λδf δΛ∂−1)+∂Λ−Λ(δf δΛΛ)++∂−1(∂Λ∂−1(δf δΛ)∗+∂Λ−Λ(∂−1(δf )∗∂Λ)+]δg dx (50)We deﬁne W (2k +1)BKP -algebra corresponding to the second Poisson bracket (50)associated with the BKP hierarchy.Its free ﬁeld realization is given by the following factorizationΛ=∂−1L ∗1L 1(51)withL 1=l j =1(∂+p j )−1k +1+2lj =1+1(∂+p j )(52)10where p j are independentﬁelds and satisfy{p i(x),p j(y)}=−δijδ (x−y)1≤i,j≤l{p i(x),p j(y)}=δijδ (x−y)l+1≤i,j≤k+1+2l{p i(x),p j(y)}=01≤i≤l,l+1≤j≤k+1+2l(53)In conclusion we have discussed the properties of the second Poisson structure associated with the matrixΨDO.These properties enable us to construct the freeﬁeld realizations for a more general class of V-algebras that correspond to the second Poisson brackets of matrixΨDO.Because of the non-locality of the V-algebras,the freeﬁeld realizations for them become more diﬃcult than for W-algebras.It would be of interest to investigate these problems for the general V-algebras.AcknowledgementsThis work was supported by the National Basic Research Project for“Non-linear Science”and Fund of CAS.References[1]A.Bilal:Non-abelian Toda theory:a completely integrable model forstrings on a black hole background,Nucl.Phys.B422,258(1994) [2]A.Bilal:Multi-component KdV hierarchy,V-algebra and non-abelianToda theory,Lett.Math.Phys.32,103(1994)[3]A.Bilal:Non-local matrix generalizations of W-algebras,Comm.Math.Phys.170,117(1995)[4]Y.Cheng:The freeﬁeld realization of W(n)KP -algebras,Lett.Math.Phys.33,159(1995)[5]Y.Cheng:Modifying the KP,the n th c onstrained KP hierarchies andtheir second Hamiltonian structures,Comm.Math.Phys.171,661 (1995)11[6]L.A.Dickey:Soliton equation and Hamiltonian systems.Advanced Seriesin Math.Phys.vol.12Singapore:World Scientiﬁc1991[7]F.Yu:Many boson realizations of universal nonlinear W∞-algebras,modiﬁed KP hierarchies,and graded Lie algebras,Lett.Math.Phys.29,175(1993)[8]J.Mas, E.Ramos:The constrained KP hierarchy and the generalisedMiura transformation,q-alg/9501009[9]E.Ramos,S.Stanciu:On the supersymmetric BKP hierarchy,Nucl.Phys.B427,338(1994)[10]E.Date,M.Jimbo,N.Kashiwara,T.Miwa:Transformation groups forsoliton equations,In:Nonlinear integrable systems-classical and quan-tum theory.M.Jimbo and T.Miwa eds.Singapore:World Scientiﬁc,198312。

Completable filiform Lie algebras

and H p (g, g) = Z p (g, g) | B p (g, g) , p ≥ 1. This quotient space is called the cohomology space of g of degree p with values in g [12]. For p = 0, then we put B 0 (g, g) = {0} and H 0 (g, g) = Z 0 (g, g). This last space can be identiﬁed to the space of all g-invariant elements that is {X ∈ g such that adY (X ) = 0 ∀Y ∈ g} . Then Z 0 (g, g) = Z (g) (the center of g). Further, we have Z 1 (g, g) = {f : g −→ g | δf = 0} . 2
Abstract We determine the solvable complete Lie algebras whose nilradical is isomorphic to a ﬁliform Lie algebra. Moreover we show that for any positive integer n there exists a solvable complete Lie algebras whose second cohomology group with values in the adjoint module has dimension at least n.
Deﬁnition 2. The dimension r (n) of a maximal torus of the preceding form is called the rank of n. The rank is an invariant of the nilpotent Lie algebra. By abuse of notation, we often say that the semidirect product n ⊕ t is of rank r , where t is a maximal torus of n and dim (t) = r . A system of generators {X1 , .., Xn } satisfying the proposition will be called a t-msg. Theorem 1. Let g be a solvable complete Lie algebra. Then 1. g decomposes as g = t + n, where n is the nilradical of g and t is a maximal toral subalgebra. 2. the restriction adth|n is an abelian subalgebra of Der (n), and a maximal torus over n. This result, whose proof can be found in [15], allows to refer to the decomposition directly, which is of interest for the structural analysis of such a Lie algebra, since it shows that we can restrict ourselves to the class of decomposable Lie algebras [1]. A nilpotent Lie algebra n will be called completable if n ⊕ t is complete, where t is a maximal torus of derivations of n. Recall that the minimal number of generators is given by the ﬁrst Betti number b1 (n) = dim Cn 1 n . We in particular obtain an interesting result for maximal rank Lie algebras, i.e., for nilpotent Lie algebras satisfying dim (t) = b1 (n): Theorem 2. Let n be a nilpotent Lie algebra of maximal rank and t a maximal torus. Then the semidirect product t ⊕ n is a solvable complete Lie algebra. This theorem, due to Zhu and Meng [13], has important consequences for the theory of complete Lie algebras. We ennounce a couple : It is easy to see that two solvable complete Lie algebras of maximal rank are isomorphic if and only if their corresponding nilradicals are isomorphic. Moreover, it can be shown that if the nilradical is split, then it is of maximal rank if and only if any component is also of maximal rank. This allows to reduce the study to the determination of complete solvable algebras whose nilradical is nonsplit. Observe the analogy with the classical theory. Remark 1. It should be remarked that there exist complete Lie algebras whose nilradical is not of maximal rank. In low dimensions they have been classiﬁed, but for the general case there only exist partial results 4

On Representations of Toroidal Lie Algebras

as scalars on irreducible module but as invertible operators. So the study of modules does not follow the standard methods of aﬃne modules and needs completely diﬀerent techniques. Surprisingly, a lot of information can be obtained on an irreducible module for toroidal Lie algebra when we assume all weight spaces are ﬁnite dimensional. For example the part of the center which acts non-trivially on an irreducible module will have a deﬁnite shape. In the sense that if we mod out some arbitrary part of the center and look for an irreducible module with ﬁnite dimensional weight spaces, they may not exist. For example if we mod out all the non-zero degree center then module does not exist with ﬁnite dimensional weight spaces and with non-zero central (zero degree) action. Further it is proved that the ratio of standard zero degree central operators on an irreducible module with ﬁnite dimensional weight spaces is rational. See ref. [E8]. The most striking result is the classiﬁcation of irreducible integrable modules for toroidal Lie algebras with ﬁnite dimensional weight spaces. But the classiﬁcation is disappointing as the modules are tensor product of evaluation modules of aﬃne Lie algebras. In particular the characters are product of aﬃne characters. See references [E4,E5 and Y]. It is worth noting that integrable modules are not completely reducible and such modules are constructed by Chari and Le Thang [CL]. So the classiﬁcation of integrable modules not necessarily irreducible may be interesting and their characters being Weyl group invariant may hold some signiﬁcance. The next natural issue is to generalize the Virasoro Lie algebra. Thus we consider the Lie algebra of diﬀeomorphisms on n dimensional torus which is known to be isomorphic to DerA the derivation algebra of A. Several attempts have been made by physicists to give a Fock space representation to DerA or to its extension (see [FR]). They all failed to produce any interesting results due to lack of proper deﬁnitions of “normal ordering” among other 3

The multiple zeta value algebra and the stable derivation algebra

w 0
new-zeta space is a ﬁltered graded Q-vector space which is a quotient of the ideal Z>2 ; N Z = Z>2 (Z>0 )2 . For more details, see §§1.3. Theorem 4.2.2 . There is a canonical surjective Q-linear map ΨDR : D∗ ։ N Z as ﬁltered graded Q-vector space, where D∗ is the ﬁltered graded dual of the stable derivation algebra. There is a standard conjecture on the structure of D which is a combination of the conjectures (§§2.4) by Y. Ihara and P. Deligne concerning the Galois represenl 1 − {0, 1, ∞}). tation on π1 (PQ Conjecture 2.4.1 . D is a free Lie algebra generated by one element in each degree m (= 3, 5, 7, · · · ). As a corollary of Theorem 4.2.2, we get the following Proposition 4.3.7 . If we assume Conjecture 2.4.1, then we get a “ﬁnal upper bound” 1 dimQ Zw dw where dw is the conjectured value of dimQ Zw in Dimension Conjecture ([23], see also §§1.2). Modulo Conjecture 2.4.1, Dimension Conjecture is equivalent to Conjecture 4.3.8 . ΨDR is an isomorphism. Particularly this conjecture claims that the dual vector space of N Z might admit a co-Lie algebra structure deduced from that of D∗ . This paper is organized as follows. The deﬁnitions and basic known facts on multiple zeta values will be reviewed and the new zeta space will be introduced in §1. §2 is devoted to reviewing the stable derivation algebra which appeared in l 1 some works by Y. Ihara on the Galois representation on π1 (PQ − {0, 1, ∞}). In §3,

A noisy system with a flattened Hamiltonian and multiple time scales

NATELLA V. O’BRYANT Department of Mathematics, University of California - Irvine 272 Multipurpose Science & Technology Building, Irvine, CA 92697

We consider a two-dimensional weakly dissipative dynamical system with time-periodic drift and diﬀusion coeﬃcients. The average of the drift is governed by a degenerate Hamiltonian whose set of critical points has an interior. The dynamics of the system is studied in the presence of three time scales. Using the martingale problem approach and separating the time scales, we average the system to show convergence to a Markov process on a stratiﬁed space. The averaging combines the deterministic time averaging of periodic coeﬃcients, and the stochastic averaging of the resulting system. The corresponding strata of the reduced space are a two-sphere, a point and a line segment. Special attention is given to the description of the domain of the limiting generator, including the analysis of the gluing conditions at the point where the strata meet. These gluing conditions, resulting from the eﬀects of the hierarchy of time scales, are similar to the conditions on the domain of skew Brownian motion and are related to the description of spider martingales. Keywords : Hamiltonian systems, Markov processes, stochastic averaging, martingale problem.

What Is Real

What Is RealChristof have created a false world for Truman, in fact we also live in a false world creating by ourselves. We have defined real and false, and we all cry for real. However, we only find ourselves go farther and farther away from real. Real has been institutionalized. We are in a real crisis.When businessmen are trying various approaches to make profits, real is more consumers and more money. When media is concentrated on making up all kinds of interesting news about superstars, real is more attention and more audience. When ladies are spending money and energy on plastic surgery, real is more attraction and more beauty. When officers are learning to embezzle and flatter, real is more power and trust. When students are getting used to preparing cheat sheet, real is more scores and more praises. When all people are engaged in pretending and disguising, real no longer exists.From the conversation between David and Teddy in AI(“Teddy, I suppose Mummy and Daddy are real, aren‟t they?”Teddy said, “you ask such silly questions, David. Nobody knows what …real‟really means.”), we can learn that though it is human beings who define real, they do not exactly what it is since people are more and more good at creating false things.However, there are still a few real sometimes in some situations. When some companies or businessmen are being discovered adopting illegal means to produce seemingly tempted products, real does exist. When some news are being proved not to be true, real does exist. When the side effects of the plastic surgery are becoming evident, real does exist. When the officers‟ illegal behaviors are revealed to the public, real does exist. When students are being caught cheating and given punishment, real does exist. When people‟s true purposes are being disclosed and no longer trusted by their good friends or leaders, real does exist.When we are young, we all believe that adults will not lie. In fact, everyone lies because everyone has secrets in some degree that he/she does not want others to know. When we grow up, we also begin to lie. In this world full of lies, real is in fact false. In this sense, maybe Christof is right, “there is no more truth out there than there is in the world I created for you. The same lies, the same deceit. But in my world, you have nothing to fear.” Outside the world,everyone has to learn to accept the fact and lie himself. Real is either changed completely or beautified. This is real and we cannot change it. Real is what we see seemingly but not know deeply.It seems that the more we are nearer to the truth, the more dangerous we will be. So keeping a blind eye and being simple is real in this world. Only by this way can we live a real life and be real.。

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∗ Les
auteurs ont ´ et´ e support´ es par le projet de recherche PR1/05-13283 de la UCM. : ancochea@mat.ucm.es ‡ e-mail : rutwig@mat.ucm.es § e-mail : lucigarcia@mat.ucm.es
1 Rappellons qu’une forme r´ eelle d’une alg` ebre de Lie complexe g est une alg` ebre r´ eelle g′ telle que g′ ⊗ C ≃ g.
† e-mail
1
Comme ad(X ) est diagonalisable et le nilradical est invariant par cet op´ erateur, on peut trouver une base {X1 , .., Xn = X } de vecteurs propres telle que {X1 , .., Xp+q } soit une base de n, {Xp+q+1 , .., Xn = X } soit une base de t et {Xp+1 , .., Xn = X } une base de ker ad(X ). Alors on consid` ere le syst` eme lin´ eaire de racines S ` a dim g − 1 variables xi dont les ´ equations sont xi + xj = xk si la composante du vecteur [Xi , Xj ] sur Xk est non nulle. De cette fa¸ con, on obtient un crit` ere de rigidit´ e: Proposition 1 [1] Si g est rigide, alors pour tout vecteur r´ egulier X on a rank(S ) = dim n − 1. En utilisant ce syst` eme, on a classiﬁ´ e les alg` ebres de Lie r´ esolubles rigides complexes jusqu’` a dimension huit [1]. De plus, cette propri´ et´ e lin´ eaire implique une autre d’ordre structurale : D´ eﬁnition 2 Une alg` ebre de Lie r sur le corps K est dite compl` etement r´ esoluble si pout tout X ∈ r les valeurs propres de adX sont des ´ el´ ements de K. En particulier, cette structure implique l’existance d’une suite r = In ⊃ In−1 ⊃ ... ⊃ I0 = 0, o` u les Ik sont des id´ eaux de codimension 1 dans Ik+1 . Proposition 2 Soit r une alg` ebre de Lie r´ esoluble rigide complexe. Alors r est compl` etement r´ esoluble. D´ emonstration. D’apr` es la proposition 1, pour toute alg` ebre de Lie r´ esoluble complexe rigide il existe une base {X1 , .., Xr } du tore ext´ erieur t telle que les valeurs propres des op´ erateurs adjointes sont des enti` eres. En eﬀet, ´ etant donn´ ee une base {Y1 , .., Yr } de t, les valeurs propres des op´ erateurs ad(Yj ) sont des solutions du syst` eme S (X ). Comme celui-ci est ` a coeﬃcient entiers, il existe une base de solutions rationelles, et pourtant on peut trouver une base de solutions enti` eres. Dans [2] on a montr´ e l’existence des alg` ebres de Lie rigides complexes nonr´ eelles. D’apr` es le th´ eor` eme du rang, c ¸a implique que le tenseur de structure du nilradical n’est pas r´ eel, c’est-` a-dire, le nilradical n’admet pas des formes r´ eelles1. On appellera une alg` ebre de Lie complexe sans formes r´ eelles purement complexe. Proposition 3 Toute alg` ebre de Lie r´ esoluble rigide complexe g = n ⊕ t dont le nilradical n’est pas purement complexe admet des formes r´ eelles. La preuve est imm´ ediate. En eﬀet, si le nilradical n’est pas purement complexe, alors les constantes de structure de n sont r´ eelles. Par le th´ eor` eme du rang, on peut trouver des g´ en´ erateurs du tore ext´ erieur ayant des valeurs propres enti` eres. Alors la restriction par scalaires donne une alg` ebre de Lie r´ eelle, appel´ ee la forme r´ eelle normale. De plus, toute autre forme r´ eelle (n′ ⊕ k) poss` ede au moins un g´ en´ erateur X ∈ k tel que ad(X ) n’est pas diagonalisable sur R, mais X ∈ t vu comme g´ en´ erateur de l’alg` ebre complexiﬁ´ ee.
J. M. Ancochea Berm´ udez† , R. Campoamor-Stursberg‡ and L. Garc´ ıa Vergnolle§ Dpto. Geometr´ ıa y Topolog´ ıa, Facultad CC. Matem´ aticas U.C.M. Plaza de Ciencias 3, E-28040 Madrid
∗
R´ esum´ e On montre qu’une alg` ebre de Lie r´ esoluble rigide r´ eelle n’est pas n´ ecessairement compl` etement r´ esoluble. On construit un exemple n ⊕ t de dimension minimale dont le tore ext´ erieur t n’est pas form´ e par des d´ erivations ad-semisimples sur R. Nous ´ etudions les formes r´ eelles des nilradicaux des alg´ ebres de r´ esolubles rigides en dimension n ≤ 7 et donnons la classiﬁcation des alg` ebres r´ esolubles rigides sur R en dimension 8.
arXiv:math/0510275v1 [math.RT] 13 Oct 2005
Les alg` ebres de Lie r´ esolubles rigides r´ eelles ne sont pas n´ ecessairement compl` etement r´ esolubles
1
பைடு நூலகம்
Preliminaries
Rappellons qu’une alg` ebre de Lie g sur un corps K de caract´ eristique 0 est dite rigide lorsque son orbite dans le sch´ ema Ln donn´ e par les r´ elations de Jacobi sur la famille des lois d’alg` ebres de Lie sur Kn est ouverte sous l’action du groupe 2 GL(n, K). Si H (g, g) est le deuxi` eme groupe de cohomologie ` a valeurs dans le module adjoint, le crit` ere de rigidit` e de Nijenhuis et Richardson [10] donne une condition suﬃsante : si H 2 (g, g) = 0, alors l’alg` ebre g est rigide dans Ln . Dans [6], on montre que sur un corps K alg´ ebriquement clos de caract´ eristique 0, toute alg` ebre de Lie r´ esoluble rigide v´ eriﬁe la d´ ecomposition suivante : g = n ⊕ t, o` u n est le nilradical de g et t est une sous-alg` ebre ab´ elienne dont les ´ el´ ements sont adg -semi-simples, appel´ e tore ext´ erieur de g. En utilisant cette propri´ et´ e, on peut d´ evelopper une th´ eorie de racines qui permet classiﬁer les alg´ ebres de Lie r´ esolubles rigides complexes par rapport au spectre du tore ext´ erieur. D´ eﬁnition 1 Un ´ el´ ement X ∈ t est dit r´ egulier si la dimension du noyau ker ad(X ) est minimale.