On Lie 2-bialgebras

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索罗干-班东甘辛学习模型教程说明书

Application Sorogan -Bandongan Model in LecturesReaction MechanismRinaningsih*, Suyatno Sutoyo Department of Chemistry Universitas Negeri SurabayaSurabaya, Indonesia ********************.idAsep Kadarohman, Harry FirmanDepartment of Chemistry Indonesia University of EducationBandung, IndonesiaAbstract — Sorogan-Bandongan is an integrated lecturemodel coming from the traditional learning that are implemented at Islamic boarding school in Indonesia. It starts with Sorogan and ends with Bandogan. The implementation of integrated lecture model of Sorogan-Bandongan with stages of students reading and doing handout assignment; diagnostic test, material explanation by lecturer; students work on worksheets (Sorrogan); class discussion on reinforcing the material (Bandongan); final test. Sorogan-Bandongan becomes a lecture model in learning reaction mechanism with the average of handout assignments 51.8; diagnostic test 47.9; final test 60.4. The correlation between those three results is 0,511. It is found that to learn reaction mechanism needed precondition configuration concept of electron, hybridization, molecular form, Lewis acid and base, and steric hindrance.Keywords— model Sorogan-Bandongan, test diagnostic, learning reaction mechanismI. I NTRODUCTIONReaction mechanism is a step-by-step description of chemical reaction event. This event of chemical reaction is an event on the molecular level that involves the outer shell of electron [1]. Tastan reveals that the difficulties in learning reaction mechanism include determinant of reaction speed and transition state [2]. These difficulties cause the lack of professionalism of the teacher in teaching reaction mechanism. Dicks has done a research by implementing several methods in lectures, so it can increase the attention [3]. Giving online assignment to the students has been done by Franz as it produced more effective lecturing method [4]. The same method is also done by Parker, working on online assignment with credit points which can improve the eagerness in learning [5]. Besides the method implementation in reaction mechanism lecture, spatial planning of classrooms can also improve performance capabilities [6]. The use of laboratories studio in S N 1 and S N 2 reaction mechanism lectures can also improve the confidence of the students [7]. The implementation of methods in reaction mechanism lectures can be done by giving assignment, classroom spatial planning, syllabus improvement and quizzes [8]. Giinersel did an integrated research to improve the involvements of the students, in this case positively correlated with their learning achievements [9, 10].The difficulties in understanding reaction mechanism, besides caused by the lack of professionalism of the teacherin teaching reaction mechanism, it is also caused by the lack of implementation of several methods in reaction mechanism lecture. In understanding reaction mechanism concept, it needs to be understood by the way of sharing between the students in a discussion activity so they can improve their skill in thinking logically. Reaction mechanism lecture should be designed to facilitate the students with opportunity to discuss and thinking logically [11]. This discovery is fulfilled in the integrated lectures models of Sorogan – Bandongan . The development of integrated lecture of Sorogan – Bandongan is a fusion of Sorogan method and Bandongan method. Sorogan is an individual learning method while Bandongan is a discussion method for material sifting by student who understands the material better to the students that have not understood. This paper will discuss the result of research about the implementation of integrated lecture of Sorogan –Bandongan model to determine the understanding of the students in learning reaction mechanism.II. M ETHODOLOGYResearch method used is Mixed Method with Embedded Experimental Design model [12]. Data were described qualitatively and then the correlation between handout performance value, diagnostic test result and final test result quantitatively were analysed. Sample used in this research are 31 students from Chemistry Department of UNESA that takes Organic Chemistry I course on even semester, academic year of 2013/2014. Data of this research are diagnostic test, handout, learning result test and questionnaire. The outline of the Sorogan –Bandongan model is as follows: students read and do assignment on the handout; diagnostic test; material explanation by the lecturer; students work on worksheets; class discussion on reinforcing the material; final test. Data analysis was done using mixed method approach [12], which was firstly done qualitatively to describe the result of diagnostic test, handout assignment, final test result and students questionnaire. Secondly, to ensure there is a correlation between diagnostic test, handout assignment and final test result, quantitative approach is applied by using Statistics [13].III. R ESULTS AND D ISCUSSIONBased on the questionnaire filled by the respondents, obtained 25 among 31 students reading the handout before attending the lecture. Out of 25 students that read the handout, 21 students stated that they understand the lectureProceedings of the Seminar Nasional Kimia - National Seminar on Chemistry (SNK 2018)easier after reading the handout. The percentage of students in understanding the material on the handout before the explanation by the lecturer is 64.0% (16 students) experienced difficulties in learning S N 1, 40.0% (10 students) S N 2, 40.0% (10 students) reaction mechanism, 12.0% (3 students) experienced difficulties in understanding how to fill the orbital, 20.0% (5 students) is unable to determine nucleophicility, and 16.0% (4 students) experienced difficulties in determining hybridization. From the result of the questionnaire it is also obtained the student's appraisal of the lecturer’s way of teaching, stated that the explanation of the lecturer was difficult to understand, because of the less precise use of language, also the material taught was too short. This is in accordance with the research done by Tastan [2]. From the result, it is emphasized that in the implementation of integrated lecture using Sorogan -Bandongan model, there must be a work of the lecturer in the form of handout and students worksheets, so only professional educators can implement this model.Chart below shows the result of diagnostic test on the difficulties in learning reaction mechanism substitutionnucleophilic.The result of diagnostic test used as consideration to step on the third syntax which is material explanation. The high percentage shows that material are increasingly difficult for the students, so that the emphasize on the material explanation is sorted according to the percentage of diagnostic test. The order of the material from the most difficult are Lewis acid and base, S N 1, electronegativity, Lewis structure, S N 2, electron configuration, hybridization and orbital filling. After the material explanation, students were asked to work on questions on the students worksheets. The lecturer tour around and give guide to the students that are having difficulties in learning individually. After finishing the worksheets individually (1) as material reinforcement, a class discussion was conducted (2). As for the final stage of the lecture, a learning result test was carried out (final test). The material difficulties on the final test can be exposed as follows: 3.0% students having difficulties in defining reaction substitution; 46.0% of the students having difficulties in understanding the concept of prerequisites; 4.5% students having difficulties on S N 1; and 56.0% of the students having difficulties to differentiate between S N 1 and S N 2.Determining the effectivity of the Sorogan – Bandongan model on the research can be done by comparing thediagnostic result of learning difficulties with material difficulties on the final test. It shows that there was drastic difficulties decline from 32.0% during the diagnostic test to 3.0% on the final test on definition of reaction substitution. Likewise, on the difficulties in learning S N 1, the diagnostic test changes from 83.0% to only 4.5% and S N 2 during diagnostic test from 48.0% to 20.0% on the final test. Prerequisite concept during the diagnostic test for Lewis acid and base is 90.0%, electron configuration 23.0%, orbital filling 13.0%, hybridization 23.0%, Lewis structure 55.0% and electronegativity 65%. If averaged, concept of prerequisite on the diagnostic test is 44.0%. That result is almost equivalent to the final test of 46.0%. The equivalent shows that the concept of prerequisite needs its own time in lecture because of the limitation of time during the lecture’s face to face interaction.Difficulties in learning S N 1 during diagnostic test and final test showed on the diagnostic test which amounted to 83.0% and final test 4.5%. The difference of 78.5% between diagnostic test and final test illustrates that this model is effective if used in learning S N 1. This illustration shows that the implementation of Sorogan – Bandongan integrated model can solve the difficulties in learning for the students. Students’ difficulties in determining the structure formula become the obstacle for the students in determining the chemical equation; when they are unable to differentiate between primary, secondary and tertiary alkyl halide, students cannot determine whether the compound undergo S N 1 or S N 2 reaction, Structure formula greatly influences the existence of steric hindrance of a compound in determining whether the compound has S N 1 or S N 2 reaction. In this research the improvised final test is very appropriate to measure the ability of the students in understanding the material of nucleophilic substitution with the integrated model of Sorogan – Bandongan . This is in accordance with the studies done by the previous researcher which methods are implemented in integrated lecture [9, 10] with the syllabus development and quizzes/test [8] could improve the learning interest [5].The decline of the average percentage from the result of diagnostic test compared with the result of the final test shows that integrated model of Sorogan – Bandongan can increase the effectivity of lectures. This discovery is in line with the previous studies, showing that the implementation of several methods in reaction mechanism lectures can improve the attention [3], involving students, centered on students [4], increasing learning interest [5], improving performance capabilities [6], and confidence [7].The discovery resulted in this research is that students find difficulties in differentiating the S N 1 and S N 2 reaction mechanism, which is caused by that students not mastering the concept of prerequisite electron configuration, hybridization, molecular form, Lewis acid and base, and steric hindrance. The percentage of difficulties in prerequisite concept, and S N 1 and S N 2 reaction mechanism are equal, amounted to 46.0% and 56.0%. The prerequisite concept still has high percentage (46.0%), even almost equals to the diagnostic test (44.0%). This is caused by the lack of time in the third step which is material explanation. Lecture duration in structured manner is only 2 credit hours (100 minutes) meanwhile to finish the prerequisite concept it can take up to 2 meetings.Prerequisite concept that has the highest percentage is Lewis acid and base with 90.0%. Lewis acid and base actually have been studied on the basic chemistry course and studied again in details in chapter I of Organic Chemistry I, which is about the Structure of Atom and Molecules. This concept is closely related with the electronegativity and chemical bond as the determiner in understanding the reaction mechanism; because the reaction mechanism is a movement of electron on the outermost shell for the occurrence of reaction with other compound. So if students do not understand the Lewis acid and base, then they will not be able to determine the reaction mechanism. Therefore the difficulties percentage of Lewis acid and base, and reaction mechanism is equal. In this research, it is found that out of 31 students as sample, only 4 students were able to work perfectly on the reaction mechanism during the final test.Based on statistical correlation between three variables (diagnostic test, handout and final test), it is obtained the correlation coefficient R of 0,511 which means there is moderate or enough relationship between diagnostic test result, reading and working on the handout, and final test result [13].IV.C ONCLUSIONS1.The learning of reaction mechanism of nucleophilicsubstitution requires the understanding of prerequisite concept of electron configuration, hybridization, molecular form, Lewis acid and base, and steric hindrance.2.The implementation of integrated model of Sorogan–Bandongan can improve lecture effectivity of nucleophilic substitution material and it is discovered that students experience difficulties in differentiating S N1 and S N2 reaction mechanism.R EFERENCES[1] Ahiakwo, Macson, J. Organic Reaction Mechanism Controversy:Pedagogical Implication for Chemical Education. AJCE , 2(2), February 2012, Pp.51-65[2] Tastan,O., Yalcinkaya, E., Boz, Y. Pre-Service Chemistry Teachers’Ideas about Reaction Mechanisme. Journal of Turkish Science Education, March 2010, Volume 7, Issue 1.[3] Dicks, P.A., Lauten, M., Koroluk, J.K., Skonieczny, T., (2012)Undergraduate Oral Examinations in a University OrganicChemistry Curriculum. Journal of Chemical Education. Published:October 18, Pp. 1506-1510[4] Franz, K., A. Organic Chemistry You Tube Writing Assignment forLarge Lecture Classes. Journal of Chemical Education. Published:2011, November 29. Pp. 497-501[5] Parker, L.L., Loudon, M.,G. Case Study Using Online Homework inUndergraduate Organic Chemistry: Result and Student Attitudes.Journal of Chemical Education. Published: 2012, November 12, Pp.37-44[6] Muthyala, S.R., Wei, W. Does Space Matter? Impact of ClassroomSpace on Student Learning in an Organic-First Curriculum. Journalof Chemical Education. Published: 2016, November 26, Pp. 45-50 [7] Collison, G..C., Cody, J., Stanford, C. An S N1-S N2 Lesson in anOrganic Chemistry Lab Using a Studio-Based Approach. Journal ofChemical Education. Published: March 21, 2012 Vol.89, Pp. 750-754[8] Aldahmash, H.A., Abraham, R.M. Kinetic Versus Static Visual forFacilitating College Students Understanding of Organic ReactionMechanisme in Chemistry. Journal of Chemical Education.Published: 2009, December, Vol.86 No. 12: Pp. 1442-1446[9] Giinersel, B.A., Fleming, A.,S. Qualitative Assessment of a 3DSimulation Program: Faculty, Students, and Bio-Organic Reaction.Journal of Chemical Education. Published: 2013, June 25, Pp. 988-994[10] Kenzie, Mc.N., Nulty, Mc.J., Leod, Mc.D., Fadden, Mc.M.,Balachandran, N. Synthesizing Novel Anthraquinone NaturalProduct-Like Compounds to Investigate Protein-Ligand Interactionin Both an in Vitro and in Vivo Assay: an Integrated Research-Based Third- Year Chemical Biology Laboratory Course. Journal ofChemical Education. Published: 2012, April 6, Pp. 743-749[11] Mercer, M.,S., Andraos, J., Jessop, G..P. Choosing the GreenestSynthesis: A Multivariate Metric Green Chemistry Exercise. Journalof Chemical Education. Published: 2011, December 5, Pp. 215-220 [12] Creswell, W. J. Educational Research (planning, Conducting, andEvaluating Quantitative and Qualitative Research) (Third edition).Canada: Pearson Education, Inc, 2008.[13] Sharma, S. Applied Multivariate Techniques. Canada: John Wiley &Sons, Inc, 1996.。

Algebra_II_Honors_Syllabus_2012-2013

Algebra 2 HonorsInstructor: Mr. Dan Chase 2012-2013 Course Syllabus Email: dchase@ Classroom: Wetmore B3Webpage: Textbook:Algebra 2, by Holt (Homework Help at/hrw.nd/gohrw_rls1/pKeywordResults?keyword=MB7+HWHelp) Course Objectives:∙Develop a variety of problem solving skills and strategies∙Work collaboratively with other students and utilize technology to research and communicate∙Foster independent learning on the part of each student∙Explore real world applications and abstract concepts of math ematics∙Prepare for success in future math courses∙Develop skills and strategies to succeed on college entrance examsCourse Topics:1. Foundations for Functions2. Linear Functions3. Linear Systems4. Quadratic Functions5. Polynomial Functions6. Exponential and Logarithmic Functions7.Rational and Radical Functions8. Conic Sections9. Introduction to MatricesMaterials:∙Calculator – Graphing calculator: TI-84 or TI-89∙Other supplies – Paper, pencil, one-subject spiral notebook, 3-ring binder (multi- or single-subject), dividers∙Computer –Many in-class activities will involve the use of a computer. It is recommended to bring your laptop to class if you have one. If not, one will be provided for use during class time.Grading Policy:Tests 50 %Quizzes 20 %Daily (homework and classwork) 20 %Comprehension Checks 10%**Homework will be assigned daily and will be checked at least once per week for grading. The grade will reflect completeness and work shown. Late work will not be accepted unless the re are extenuating circumstances and the instructor has been notified via email of these circumstances at least 24 hours in advance.Make-up Policy:According to the school handbook, students have up to one week to make up any work missed when an excused absence has occurred. It is the responsibility of the student to find out what assignments have been missed. No make-up work will be allowed for unexcused absences.Classroom Policies:1.The classroom is our place of business. Treat the people and the space around you accordingly. Be polite, respectful,engaged, and cooperative with your fellow students, teachers, and visitors.2.Be prepared for class every day with clean paper, sharpened pencil, functioning calculator, and your algebra textbook.3.Be seated in your desk when class begins and immediately follow the day’s posted agenda.4.ALL rules in your student handbook apply in the classroom including being on time; following the dress code; nothaving gum, food, or drink in classroom; etc. (Note: Bottled water is allowed in class.)Help Sessions:I will be available in my classroom most school days during the Extra Help sessions in the mornings (8:05-8:30 every day except Wednesdays when Extra Help is 8:55-9:20). I am also available while I am on duty in West House during evening study hall most Wednesday nights. Students are welcome to come by and work with me on algebra during this time, but please make arrangements with me ahead of time. Students can also arrange other times for extra help by communicating face-to-face or via email.Please make the effort to get extra help outside class when you need it. Do not wait for the day of a test or quiz!。

Generalized WDVV equations for B_r and C_r pure N=2 Super-Yang-Mills theory

a r X i v :h e p -t h /0102190v 1 27 F eb 2001Generalized WDVV equations for B r and C r pure N=2Super-Yang-Mills theoryL.K.Hoevenaars,R.MartiniAbstractA proof that the prepotential for pure N=2Super-Yang-Mills theory associated with Lie algebrasB r andC r satisﬁes the generalized WDVV (Witten-Dijkgraaf-Verlinde-Verlinde)system was given by Marshakov,Mironov and Morozov.Among other things,they use an associative algebra of holomorphic diﬀter Ito and Yang used a diﬀerent approach to try to accomplish the same result,but they encountered objects of which it is unclear whether they form structure constants of an associative algebra.We show by explicit calculation that these objects are none other than the structure constants of the algebra of holomorphic diﬀerentials.1IntroductionIn 1994,Seiberg and Witten [1]solved the low energy behaviour of pure N=2Super-Yang-Mills theory by giving the solution of the prepotential F .The essential ingredients in their construction are a family of Riemann surfaces Σ,a meromorphic diﬀerential λSW on it and the deﬁnition of the prepotential in terms of period integrals of λSWa i =A iλSW ∂F∂a i ∂a j ∂a k .Moreover,it was shown that the full prepotential for simple Lie algebras of type A,B,C,D [8]andtype E [9]and F [10]satisﬁes this generalized WDVV system 1.The approach used by Ito and Yang in [9]diﬀers from the other two,due to the type of associative algebra that is being used:they use the Landau-Ginzburg chiral ring while the others use an algebra of holomorphic diﬀerentials.For the A,D,E cases this diﬀerence in approach is negligible since the two diﬀerent types of algebras are isomorphic.For the Lie algebras of B,C type this is not the case and this leads to some problems.The present article deals with these problems and shows that the proper algebra to use is the onesuggested in[8].A survey of these matters,as well as the results of the present paper can be found in the internal publication[11].This paper is outlined as follows:in theﬁrst section we will review Ito and Yang’s method for the A,D,E Lie algebras.In the second section their approach to B,C Lie algebras is discussed. Finally in section three we show that Ito and Yang’s construction naturally leads to the algebra of holomorphic diﬀerentials used in[8].2A review of the simply laced caseIn this section,we will describe the proof in[9]that the prepotential of4-dimensional pure N=2 SYM theory with Lie algebra of simply laced(ADE)type satisﬁes the generalized WDVV system. The Seiberg-Witten data[1],[12],[13]consists of:•a family of Riemann surfacesΣof genus g given byz+µz(2.2)and has the property that∂λSW∂a i is symmetric.This implies that F j can be thought of as agradient,which leads to the followingDeﬁnition1The prepotential is a function F(a1,...,a r)such thatF j=∂FDeﬁnition2Let f:C r→C,then the generalized WDVV system[4],[5]for f isf i K−1f j=f j K−1f i∀i,j∈{1,...,r}(2.5) where the f i are matrices with entries∂3f(a1,...,a r)(f i)jk=The rest of the proof deals with a discussion of the conditions1-3.It is well-known[14]that the right hand side of(2.1)equals the Landau-Ginzburg superpotential associated with the cor-∂W responding Lie ing this connection,we can deﬁne the primaryﬁeldsφi(u):=−∂x (2.10)Instead of using the u i as coordinates on the part of the moduli space we’re interested in,we want to use the a i .For the chiral ring this implies that in the new coordinates(−∂W∂a j)=∂u x∂a jC z xy (u )∂a k∂a k )mod(∂W∂x)(2.11)which again is an associative algebra,but with diﬀerent structure constants C k ij (a )=C k ij(u ).This is the algebra we will use in the rest of the proof.For the relation(2.7)weturn to another aspect of Landau-Ginzburg theory:the Picard-Fuchs equations (see e.g [15]and references therein).These form a coupled set of ﬁrst order partial diﬀerential equations which express how the integrals of holomorphic diﬀerentials over homology cycles of a Riemann surface in a family depend on the moduli.Deﬁnition 6Flat coordinates of the Landau-Ginzburg theory are a set of coordinates {t i }on mod-uli space such that∂2W∂x(2.12)where Q ij is given byφi (t )φj (t )=C kij (t )φk (t )+Q ij∂W∂t iΓ∂λsw∂t kΓ∂λsw∂a iΓ∂λsw∂a lΓ∂λsw∂t r(2.15)Taking Γ=B k we getF ijk =C lij (a )K kl(2.16)which is the intended relation (2.7).The only thing that is left to do,is to prove that K kl =∂a mIn conclusion,the most important ingredients in the proof are the chiral ring and the Picard-Fuchs equations.In the following sections we will show that in the case of B r ,C r Lie algebras,the Picard-Fuchs equations can still play an important role,but the chiral ring should be replaced by the algebra of holomorphic diﬀerentials considered by the authors of [8].These algebras are isomorphic to the chiral rings in the ADE cases,but not for Lie algebras B r ,C r .3Ito&Yang’s approach to B r and C rIn this section,we discuss the attempt made in[9]to generalizethe contentsof the previoussection to the Lie algebras B r,C r.We will discuss only B r since the situation for C r is completely analogous.The Riemann surfaces are given byz+µx(3.1)where W BC is the Landau-Ginzburg superpotential associated with the theory of type BC.From the superpotential we again construct the chiral ring inﬂat coordinates whereφi(t):=−∂W BC∂x (3.2)However,the fact that the right-hand side of(3.1)does not equal the superpotential is reﬂected by the Picard-Fuchs equations,which no longer relate the third order derivatives of F with the structure constants C k ij(a).Instead,they readF ijk=˜C l ij(a)K kl(3.3) where K kl=∂a m2r−1˜C knl(t).(3.4)The D l ij are deﬁned byQ ij=xD l ijφl(3.5)and we switched from˜C k ij(a)to˜C k ij(t)in order to compare these with the structure constants C k ij(t). At this point,it is unknown2whether the˜C k ij(t)(and therefore the˜C k ij(a))are structure constants of an associative algebra.This issue will be resolved in the next section.4The identiﬁcation of the structure constantsThe method of proof that is being used in[8]for the B r,C r case also involves an associative algebra. However,theirs is an algebra of holomorphic diﬀerentials which is isomorphic toφi(t)φj(t)=γk ij(t)φk(t)mod(x∂W BC2Except for rank3and4,for which explicit calculations of˜C kij(t)were made in[9]we will rewrite it in such a way that it becomes of the formφi(t)φj(t)=rk=1 C k ij(t)φk(t)+P ij[x∂x W BC−W BC](4.3)As aﬁrst step,we use(3.4):φiφj= Ci·−→φ+D i·−→φx∂x W BC j= C i−D i·r n=12nt n2r−1 C n·−→φ+D i·−→φx∂x W BCj(4.4)The notation −→φstands for the vector with componentsφk and we used a matrix notation for thestructure constants.The proof becomes somewhat technical,so let usﬁrst give a general outline of it.The strategy will be to get rid of the second term of(4.4)by cancelling it with part of the third term,since we want an algebra in which theﬁrst term gives the structure constants.For this cancelling we’ll use equation(3.4)in combination with the following relation which expresses the fact that W BC is a graded functionx ∂W BC∂t n=2rW BC(4.5)Cancelling is possible at the expense of introducing yet another term which then has to be canceled etcetera.This recursive process does come to an end however,and by performing it we automatically calculate modulo x∂x W BC−W BC instead of x∂x W BC.We rewrite(4.4)by splitting up the third term and rewriting one part of it using(4.5):D i·−→φx∂x W BC j= −12r−1 D i·−→φx∂x W BC j= −D i2r−1·−→φx∂x W BC j(4.6) Now we use(4.2)to work out the productφkφn and the result is:φiφj= C i·−→φ−D i2r−1·r n=12nt n D n·−→φx∂x W BC j +2rD i2r−1·rn=12nt n −D n·r m=12mt m2r−1[x∂x W BC−W BC]j(4.8)Note that by cancelling the one term,we automatically calculate modulo x∂x W BC −W BC .The expression between brackets in the ﬁrst line seems to spoil our achievement but it doesn’t:until now we rewrote−D i ·r n =12nt n 2r −1C m ·−→φ+D n ·−→φx∂x W BCj(4.10)This is a recursive process.If it stops at some point,then we get a multiplication structureφi φj =r k =1C k ij φk +P ij (x∂x W BC −W BC )(4.11)for some polynomial P ij and the theorem is proven.To see that the process indeed stops,we referto the lemma below.xby φk ,we have shown that D i is nilpotent sinceit is strictly upper triangular.Sincedeg (φk )=2r −2k(4.13)we ﬁnd that indeed for j ≥k the degree of φk is bigger than the degree ofQ ij5Conclusions and outlookIn this letter we have shown that the unknown quantities ˜C k ijof[9]are none other than the structure constants of the algebra of holomorphic diﬀerentials introduced in [8].Therefore this is the algebra that should be used,and not the Landau-Ginzburg chiral ring.However,the connection with Landau-Ginzburg can still be very useful since the Picard-Fuchs equations may serve as an alternative to the residue formulas considered in [8].References[1]N.Seiberg and E.Witten,Nucl.Phys.B426,19(1994),hep-th/9407087.[2]E.Witten,Two-dimensional gravity and intersection theory on moduli space,in Surveysin diﬀerential geometry(Cambridge,MA,1990),pp.243–310,Lehigh Univ.,Bethlehem,PA, 1991.[3]R.Dijkgraaf,H.Verlinde,and E.Verlinde,Nucl.Phys.B352,59(1991).[4]G.Bonelli and M.Matone,Phys.Rev.Lett.77,4712(1996),hep-th/9605090.[5]A.Marshakov,A.Mironov,and A.Morozov,Phys.Lett.B389,43(1996),hep-th/9607109.[6]R.Martini and P.K.H.Gragert,J.Nonlinear Math.Phys.6,1(1999).[7]A.P.Veselov,Phys.Lett.A261,297(1999),hep-th/9902142.[8]A.Marshakov,A.Mironov,and A.Morozov,Int.J.Mod.Phys.A15,1157(2000),hep-th/9701123.[9]K.Ito and S.-K.Yang,Phys.Lett.B433,56(1998),hep-th/9803126.[10]L.K.Hoevenaars,P.H.M.Kersten,and R.Martini,(2000),hep-th/0012133.[11]L.K.Hoevenaars and R.Martini,(2000),int.publ.1529,www.math.utwente.nl/publications.[12]A.Gorsky,I.Krichever,A.Marshakov,A.Mironov,and A.Morozov,Phys.Lett.B355,466(1995),hep-th/9505035.[13]E.Martinec and N.Warner,Nucl.Phys.B459,97(1996),hep-th/9509161.[14]A.Klemm,W.Lerche,S.Yankielowicz,and S.Theisen,Phys.Lett.B344,169(1995),hep-th/9411048.[15]W.Lerche,D.J.Smit,and N.P.Warner,Nucl.Phys.B372,87(1992),hep-th/9108013.[16]K.Ito and S.-K.Yang,Phys.Lett.B415,45(1997),hep-th/9708017.。

differential algebras

Discrete Mathematics and Theoretical Computer Science1,1997,229–237On the bialgebra of functional graphs and differential algebrasMaurice GinocchioLaboratoire de Physique Th´e orique et Math´e matique,Universit´e Paris7,Tour Centrale-3`e me´e tage,2,place Jussieu, F-75251Paris Cedex05,FranceE-Mail:mag@ccr.jussieu.fr1IntroductionWe have already described the expansion of∆Σλi∂i,i.e.the powers of a Lie operator in any dimension, in order toﬁnd the expression of theﬂow of formal nonlinear evolution equations[1–3].In the one-dimensional case,the explicit expansion can be foundﬁrst in Comtet[4],and other aspects connected to the ordinary differential equations can be found in Leroux and Viennot[5]and Bergeron and Reutenauer [6].On the other hand,Grossman and Larson[7]introduced several Hopf algebras[8–10]of forests of rooted labeled trees to express the product ofﬁnite dimensional vectorﬁelds.In this paper we concentrate us on the bialgebra G of functional graphs,i.e.graphs representing mappings ofﬁnite sets in themselves [11–15].We give only the results without proofs.In a forthcoming paper[16],we develop Hopf algebra structures,computing the antipode and giving detailed aspects and proofs.In Sect.1we consider a bialgebra structure on G and three interesting subalgebras:T the set of labeled forests;S the set of permutation graphs;and L the set of well labeled forests,i.e.with strictly decreasing labels on the chains toward the roots.Recall that the graded bialgebra L is sufﬁcient for the calculus of the powers of one derivation[1],and it is extendable in a Hopf algebra,the element of which is known in the computer literature as‘heap ordered trees’.This bialgebra is useful to compute products of derivations or to transform differential monomials in differential algebras[17],and it is interesting to note that the elements L n(n edges)can be coded by the words(monomials)of the expansion of Q nq0q0q1q0q1q n1,where Q0q0q1is a noncommutative alphabet.We describe in particular the bialgebra L,ﬁrst in the polynomial form by the‘factorial’monoid L0L0n n0,where L0n is the set of words in the expansion of Q n,and second,we establish the bijective correspondence between 1365–8050c1997Chapman&Hall230M.Ginocchio L and L.We show that the calculus are easier with L,and that the product on L can be expressed in a very natural way.For example,q0n Q n,hence the(exponential)generating function of all the elements of L.We describe principally the formalism in the general case G,and the calculus uses theﬁelds F201 as well as characteristic zeroﬁelds K.In Sect.2,we describe the link with the graded differential algebra K U r0K U r and the graded algebra of differential operators K U D r0K U r D r,where U u1u2uββ1uβαα0β1is a set of indeterminates,D∂0∂1and the differential indeterminates uβασ1σp∂σ1∂σpuβαgenerate K U r[17].This shows that the above Q-calculus,which is a kind of‘dissection’on functional graphs can be used as pre-calculus in differential algebras as well as in discrete dynamical systems[18].2Bialgebra Based on the Semi-group of Functional Graphs2.1T ypes of Functional GraphsIn this paper,a connected functional graph will be called excycle[13,15].In the area of discrete dynamical systems,an excycle is known as a basin of attraction.Consider several graded andﬁltered sets of labeled functional graphs(i)E(resp.G)the set of excycles(resp.functional graphs)and designated by G n(resp.G n),the set offunctional graphs having(resp.having at most)n1nodes for n0(ii)R(resp.T)the set of labeled arborescences(resp.forests).(iii)C(resp.S)the set of cycles(resp.permutation graphs).(iv)A(resp.L)the set of well labeled arborescences(resp.forests),i.e.with strictly decreasing labels on the chains toward the root(s).As in(i),we consider for(ii)–(iv)graduations andﬁltrations.2.2Free Representation by Q-polynomialsLet G n be the semigroup of mappings of12n in itself(‘Semigroup of endofunctions’), Card G n n n and the subsemigroups,T n f;f G n f n f n1(i.e.f acyclic and Card T n n1n1,S n the symmetric group and Card S n n!L n f;f G n f i i(i.e.f subdiagonal and Card L n n!.We have the well known bijections F F:G n G n T n T n S n S n L n L n.Let Q q0q1be a noncommutative alphabet,Q0q0Q with q0noncommuting with the q i’s,Q n q1q2q n Q0n q0Q n and Q(resp.Q0),the corresponding free monoids.Taking F201as theﬁeld,consider(i)the G n module F2Q n by the F2linear incidence matrix action of f G n as l f q i q f i hence l f l gl f g.On the bialgebra of functional graphs231 (ii)the generating monomial associated with f.By morphism extension,denoted again by l f,we deﬁneQ f q f1q f2q f n l f Qιn1where Qιn q1q2q n is associated with the identityιn of G n and Qι01One again has l f l g l f g.For the following we consider(iii)The graded subsets of Q as G G n n0T T n n0S S n n0L L n n0respectively associated with G,T,S and L,with G0T0S0L01(iv)The corresponding graded F2-modules:F2G F2T F2S F2L admit components of degree n which are,respectively,G n T n S n L n modules,withdimF2G n n n dimF2T n n1n1dimF2S n dimF2L n n!(v)We will denote by R n one of the above subsemi-groups of G n(or of another category). Similarly,let R R n n0resp F2R n0F2R n be the corresponding graded subsets of Q(resp. graded F2-modules ofF2G n0F2G n.2.3Virtual Root and External ProductLet f G n I0be the set ofﬁxed points of f and H0a subset of I0,and set p q r;p r q if p q and/0otherwise.Deﬁne f0:1n0n such that f0i f i if i H0and f0i0if i H0The‘0’is the label of a virtual root added to the graph representation of f,and we will say that H0is‘conﬁned in0’,which is a ﬁxed point of f0We call‘extended endofunctions’such functions f0,denote by G0n0n1n their set, and we consider G n as a subset of G0n Similarly,we will have T0n T n S0n S n L0n L n Consequently, adding q0,we get the extended graded sets G0G0n n0the extended graded F2-module F2G0 n0F2G0n and their substructures F2T0F2S0F2L0Now letφG0mχ0be the characteristic function of H0φ10,and writeQφlφQιm qφ1qφ2qφmm ∏i1qφi(cf.Figures1and2).WithψG0n,consider the F2-bilinear product in F2G0deﬁned byQφQψQψm∏i1qφi nχ0i q0q1q n12On the right-hand side we have a sum of concatened monomials,and on the right factor the substitutions q0q0q1q n and q h q h n when h0232M.Ginocchio On the other hand,the product belongs to F2G0m n This external product is associated with unit1and F2G0is‘.’graded.To see this consider i j k being0three homogeneous polynomials,A A q0;q i F2G0mB B q0;q j F2G0nC C q0;q k F2G0pthen by(2)A B B q0;q j A q0q1q n;q i n3 and so,using deg B C n p,A q0;q iB q0;q jC B q0;q j A q0q1q n;q i n CCB q0q1q p;q j p A q0q1q p q1p q n p;q i n pA q0;q i CB q0q1q p;q j pA q0;q iB q0;q j CMoreover,because T n S n L n are subsemi-groups of G n one can see that F2R0F2T0F2S0F2L0are‘.’graded subalgebras of F2G0HenceProposition1Let the sequence G0m m1of the sets of the extended endofunctions in12m and Q0q0q1be a noncommutative alphabet.ForφG0m let Qφ∏m i1qφi be the generating monomial ofφand the graded module F2-module F2G0n0F2G0n on F201generated by all the φsThen F2G0is a graded algebra for the associative product with unit1QφQψQψm∏i1qφi nχ0i q0q1q n1whereψG0n andχ0is the characteristic function ofφ10Moreover,if R0m m1is a sequence of subsets associated with subsemi-groups of the sequence G0m m1, then F2R0n0F2R0n is a graded subalgebra of F2G02.4Splitting Operatorδn F2G0This operator substitutes the n-coproduct∆n of the Leibniz–Lie type.Associate to A Q0the left linear operatorτn A acting on B Q0,such that,if A G0m B G0n,then Bτn A BA if degB n,and0 otherwise,where BA is the concatenation of B and A.(i)Now let f G m and H0as in Sect.3,and noteﬁrst that ifτn is viewed as acting on f,then for i1m one hasτn f i n f i n,and by f0i ¯χ0i f i one hasτn f0i n¯χ0i f0i n,where¯χ01χ0According to(2),deﬁne forφG0mδn Qφτnm∏i1qφi nχ0i q0q1q n14If d0Card H0the expansion(4)gives a sum of n1d0generating monomials of functionsψκof n1n m into0n1n m,and the corresponding functional graphs factorized in commutative excycles.On the bialgebra of functional graphs233 The operatorδn A is left linear on F2G0,and(2)can be writtenQφQψQψδn Qφ5 (ii)Moreover,δp is a graded antimorphism for‘’δp A Bδp Bδp n A6 where n degB and p N.For this to compute with(5)and A B C as in Sect.3,Cδp A B A B C A B C B Cδp n A Cδp Bδp n A.If p0we recover A B Bδn A and Bδk A0if k degB(iii)Also,δn is a powerδnδnδδ1δ017 For this to compute,δpδn A q0;q iδpτn A q0q1q n;q i nτn p A q0q1q p q1pq n p;q i n pδn p A q0;q i.(iv)Deﬁne the left linear operatorµin F2G0by the expansionµ∑n0δn8By left linear action ofµA on F2G0,we get A B BµA for A B F2G0with the antimorphism propertyµA BµBµA9 which express the associativity of‘’.Proposition2Let A F2G0m B F2G0n Then the splitting linear operatorδp deﬁned left linearly by Bδp A A B if p=n,and0otherwise,veriﬁesδpδp withδδ1δ01andδp A Bδp Bδp n A Moreover,µ∑n0δn is an antimorphism in F2G0such that A B BµA2.5Exponential Generating Function of the Monomials of L0All the words of L0n(i.e.subdiagonals)are obtained from the expansion of Q n q0q0q1q0q1 q n1F2L0and Q01By equation(3),one has Q m Q n Q m n,and if A F2L0m B F2L0n we have A B F2L0m n,and then we recover that F2L0is stable for the product‘’.Because Q1q0,the associativity givesQ n q0n10 With the Q[[t]]-modules on L0,one has the exponential generating functionexp tq0∑n0t nn!Q n11exp sq0exp tq0exp s t q0234M.Ginocchio2.6ExamplesConsider equations (4)and (5)for Q ψq n 0.2.6.1Rooted T rees with n=1δq 20q 1τq 0q 12q 2τq 0q 0q 2τq 0q 1q 2τq 1q 0q 2τq 1q 1q 2(Figure 3)q 20q 1q 0q 0q 0q 12q 2q 30q 2q 20q 1q 2q 0q 1q 0q 2q 0q 1q 1q 2(Figure 4)2.6.2Excycles with n=2δ2q 23q 1q 0τ2q 25q 3q 0q 1q 2τ2q 25q 3q 0τ2q 25q 3q 1τ2q 25q 3q 2(Figure 5)q 23q 1q 0q 20q 20q 25q 3q 0q 1q 2q 20q 25q 3q 0q 20q 25q 3q 1q 20q 25q 3q 2(Figure 6)3Differential Algebra3.1Differential indeterminatesLet D ∂0∂1where ∂α∂∂ξαthe αth canonical derivation in S K ξthe algebra of formal power series in ξξ0ξ1,where K is a characteristic zero ﬁeld.If S N N is the set U u 1u 2u ββ1u αβα0β1with u αβS consider U as a set of indeterminates,u αβσ1σp ∂σ1∂σp u αβasdifferential indeterminates,replace S N N by KU ,and consider the graded differential algebra K Ur 0K U r and the graded algebra of differential operators K U D r 0K U r D r.To each W F 2R 0we associate the differential operator W U U D ;for example,with W r U K U r one hasW UW 0UW 1Uα∂αW 2Uαβ∂α∂βW 0U∑r 1W r U D r12We will use now the summation convention.3.2Brackets in K UDeﬁne for u v wU the multilinear operations valued in K U .3.2.1Arborescent Brackets (Valued in K U 1)u v uv w u vβu αv βα,henceu v Du αv βα∂β(1ﬁxed point sent to ‘0’)uv wγu αv βw γαβ,henceu v Du αv βw γαβ∂γ13Also,for AK UrBK UsA Bβ1βsA α1αr B β1βsα1αr3.2.2Circular Brackets (Valued in K U 0)uu αα(1ﬁxed point),u vu ααv ββ(2ﬁxed points)u vu αβv βα2cycleu v wu αγv βαw γβ3cycle14On the bialgebra of functional graphs 2353.2.3Mixed Brackets (Valued in K U 0)Let E be a proper excycle (i.e.with no ﬁxed point);we can write it EA i 1A i 2A i p ,where the A i k ’s are arborescences with root i k If in each arborescence A i k is reduced to its root i k ,we recover simply acycle Ei 1i 2i p Now let F k be the forest under i k ,i.e.obtained by cutting the root of A i k ,and deﬁned with F i k U F i k u j ;j N i k ,where N i k is the set of nodes of F i k :E Uu F i 1i 1u F i 2i 2u F i pi pF i 1U u i 1α1αp F i 2U u i 2α2α1F i p U u i p αp αp13.3Action of F 2R 0Moreover,F 2R 0operates K -linearly in K U with values in K U D .For this let φG 0m H 0φ1for j 0m I 1m ,and H u β1u β2u βm U ,a word on U of length m .Then the action isQ φ∏i Iq φiQ φH∏j I∏i H j∂αiu αj βj∏k H 0∂αk15The differential monomial Q φH is such that u βj is associated with j in the domain I of φIf d j is the degree in q j (in-degree of the node labeled by ‘j ’),then u αj βj is derived d j times and the indices of derivation are related to the places of the q j ’s in the word.Similarly,the differential operator D r is characterized by the number r (degree of the root)of the q 0’s and their places.So we can summarize:In a word A R 0where q j is at the place (i),then in A H the j th letter of H is derived according to i,i.e.∂αi acts.One has,in particular,taking H u 1u 2:Arborescent brackets 1U 1q 0U u 1α1∂α1u 1Dq 0q 0U u 1α1u 2α2∂α1∂α2u 1u 2D 2q 0q 1U u 1α1α2u 2α2∂α1u 2u 1D q 3q 3q 0U u 1α1u 2α2u 3α3α1α2∂α3u 1u 2u 3Dq 0q 0q 2q 2U u 1α1u 2α2α3α4u α33u α44∂α1∂α2u 1u 3u 4u 2D 2Circular brackets q 1U u 1α1α1u 1q 1q 2U u 1α1α1u 2α2α2u 1u 2q 2q 1U u 1α1α2u 2α2α1u 1u 2q 3q 1q 2U u 1α1α3u 2α2α1u 3α3α2u 1u 2u 33.4Product of Differential OperatorsThe product (2)on words with correspondence (15)gives the product of differential operators.We state,without proof,Proposition 3Let the graded differential algebra K U r 0K U r and the graded algebra of differ-ential operators K U D r 0K U r D r Let φG 0m I 1m H j φ1j for j 0m and H u β1u β2u βm a word on U of length m.Then the mapping of F 2G 0into K U D which associates to the generating monomial Q φ∏i I q φi of φthe differential operator Q φH ∏j I ∏i H j ∂αi u αj βj ∏k H 0∂αk236M.Ginocchio is a morphism,such that ifψG0n and K is a word on U of length n,one has QφH QψK QφQψKH, where KH is the concatenation of K and H.ExampleA q0B q2q1q0H u4K u1u2u3A B q2q1q0q0q1q2q3q2q1q0q0q2q1q0q1q2q1q0q2q2q1q0q3(Figure7)A H u1DB K u1u2u3DA HB K u1u2u3u4D2u4u1u2u3D u1u2u4u3DObserve that:u4u1u2u3D u1α1α2α4u2α2α1u3α3u4α4∂α3u1α1α2u2α2α1α4u3α3u4α4∂α3which corresponds to q2q1q0q1q2,i.e.the second and third terms in the graph expansion. AppendixTo view Figures1–7,click here.To return to the main paper,click on the red box.References[1]Ginocchio,M.(1995).Universal expansion of the powers of a derivation,Letters in Math.Phys.34(4),343–364.[2]Ginocchio,M.and Irac-Astaud,M.(1985).A recursive linearization process for evolution equations.Reports on Math.Phys.21,245–265.[3]Steeb,W.H.and Euler,N.(1988).Nonlinear Evolution Equations and Painlev´e Test.World Scien-tiﬁc.[4]Comtet,L.(1973).Une formule explicite pour les puissances successives de l’op´e rateur de d´e rivationde m.Roy.Acad.Sci.276A,165–168.[5]Leroux,P.and Viennot,G.(1986).Combinatorial resolution of systems of differential equations I:ordinary differential equations.Actes du colloque de combinatoire´e num´e rative,Montr´e al.Lecture Notes in Mathematics1234,pp.210–245.Springer-V erlag.[6]Bergeron, F.and Reutenauer, C.(1987).Une interpr´e tation combinatoire des puissances d’unop´e rateur diff´e rentiel lin´e aire.Ann.Sci.Math.Quebec11,269–278.[7]Grossman,R.and Larson,R.G.(1989).Hopf-algebraic structures of families of trees.J.Algebra126,184–210.[8]Joni,A.A.and Rota,G.C.(1979).Coalgebras and bialgebras in combinatorics.Studies.in Appl.Math.61,93–139.On the bialgebra of functional graphs237 [9]Nichols,W.and Sweedler,M.E.(1980).Hopf algebras and combinatorics,in‘Umbral calculus andHopf algebras’.Contemp.Math.6.[10]Sweedler,M.E.(1969).Hopf Algebras.Benjamin.[11]Berge,C.(1983).Graphes.Gauthier-Villars.[12]Comtet,L.(1974).Advanced Combinatorics.Reidel.[13]Denes,J.(1968).On transformations,transformation-semigroups and graphs.In Erd¨o s-Katona,ed-itor,Theory of Graphs.Academic Press,pp.65–75.[14]Foata,D.and Fuchs,A.(1970).R´e arrangements de fonctions et d´e b.Theory8,361–375.[15]Harary,F.(1959).The number of functional digraphs.Math.Annalen138,203–210.[16]Ginocchio,M.On the Hopf algebra of functional graphs and differential algebras.Discr.Math.Toappear.[17]Kaplansky,I.(1976).Introduction to Differential Algebras.Springer-V erlag.[18]Robert,F.(1995).Les syst`e mes Dynamiques Discrets.Springer-V erlag.。

一类超Heisenberg-Virasoro代数的超双代数结构

( ) ( ) ϕ Li , Lj ⊂ Li+ j ， Ker (1−τ ) ⊂ Kerϕ ， ϕ ⋅(1⊗ϕ ) ⋅ 1+ ξ + ξ 2 =0 。
定义 2.2：李超余代数 ( L, ∆) 由超向量空间 L 和线性映射 ∆ : L → L ⊗ L 构成，且满足
( ) ( ) ∆ Li ⊂ ∑ Lj ⊗ Lk ， Im ∆ ⊂ Im (1−τ ) ， 1+ ξ + ξ 2 ⋅(1⊗ ∆) ⋅ ∆ =0 。
c(r) = 0
(2.3)
DOI: 10.12677/pm.2019.97102
784
理论数学
李美君
3) 元素 r ∈ Im (1−τ ) ⊂ L ⊗ L 称为满足修正的 Yang-Baxter 方程，如果
x ⋅c(r ) = 0 ， ∀x ∈ L
(2.4)
文献[1]给出了以下两个结果，元素 r 满足(2.3)当且仅当满足(2.4)。设 L 是李超代数，且
((2
−
n)
λ
+
(n
−1)η
)
In
⊗
I0
+

(n
−
2)
λ
+
2
− 2
nη
+
n 2
ρ

I0
⊗
In
.
(2.6)
( L0 ) ≡ 0 表示 ( L0 ) ≡ 0(mod ( I0 ⊗ I0 )) ，即 ( L0 ) ∈ ( I0 ⊗ I0 ) 。
3. 主要结果及证明过程
本文的主要结果可表述为以下两个定理。
∑ [ x, ai ] ⊗ bi + (−1) x ai ai ⊗[ x,bi ]

Combinatorial Hopf algebras and Towers of Algebras

a rXiv:071.3744v1[mat h.CO ]19Oct27COMBINATORIAL HOPF ALGEBRAS AND TOWERS OF ALGEBRAS NANTEL BERGERON,THOMAS LAM,AND HUILAN LI Abstract.Bergeron and Li have introduced a set of axioms which guarantee that the Grothendieck groups of a tower of algebras L n ≥0A n can be endowed with the structure of graded dual Hopf algebras.Hivert and Nzeutzhap,and independently Lam and Shimozono constructed dual graded graphs from prim-itive elements in Hopf algebras.In this paper we apply the composition of these constructions to towers of algebras.We show that if a tower L n ≥0A n gives rise to graded dual Hopf algebras then we must have dim(A n )=r n n !where r =dim(A 1). 1.Introduction This paper is concerned with the interplay between towers of associative alge-bras,pairs of dual combinatorial Hopf algebras,and dual graded graphs.Our point of departure is the study of the composition of two constructions:(1)the construc-tion of dual Hopf algebras from towers of algebras satisfying some axioms,due to Bergeron and Li [3];and (2)the construction of dual graded graphs from primitive elements in dual Hopf algebras,discovered independently by Hivert and Nzeutchap [8],and Lam and Shimozono:tower of algebras −→combinatorial Hopf algebra −→dual graded graph (1.1)The notion of a pair (Γ,Γ′)dual graded graphs is likely to be the least familiar.They were introduced by Fomin [5](see also [13])to encode the enumerative prop-erties of the Robinson-Schensted correspondence and its generalizations.The ﬁrst arrow in (1.1)is obtained by using induction and restriction on the Grothendieck groups.The second arrow is obtained by using (some of the)structure constants of a combinatorial Hopf algebra as edge multiplicities for a graph.We review these constructions in Sections 2and 3.The notion of combinatorial Hopf algebra usedhere is related,but slightly diﬀerent from the one in [1].The key example of all three classes of objects arises from the theory of symmet-ric functions.In [6],L.Geissinger showed that the ring Sym of symmetric functions is a graded self-dual Hopf ing the work of Frobenius and Schur,Zelevin-sky [14]interpreted the Hopf structurein terms of the Grothendieck groups of thetower of symmetric group algebras n ≥0C S n .Finally it follows from the classical work of Young that the branching rule for the symmetric group,or equivalently2NANTEL BERGERON,THOMAS LAM,AND HUILAN LIthe Pieri rule for symmetric functions,gives rise to the Young graph on the set of partitions.The Young graph is the motivating example of dual graded graphs.In recent years it has been shown that other graded dual Hopf algebras can be obtained from towers of algebras.In[11]Malvenuto and Reutenauer establish the duality between the Hopf algebra NSym of noncommutative symmetric functionsand the Hopf algebra QSym of quasi-symmetric functions.Krob and Thibon[9] then showed that this duality can be interpreted as the duality of the Grothendieckgroups associated with n≥0H n(0)the tower of Hecke algebras at q=0.For more examples,see[2,7,12].It is very tempting,as suggested by J.Y.Thibon,to classify all combinato-rial Hopf algebras which arise as Grothendieck groups associated with a tower of algebras n≥0A n.The list of axioms given by theﬁrst and last author in[3]guar-antees that the Grothendieck groups of a tower of algebras form a pair of graded dual Hopf algebras.This list of axioms is not totally satisfactory as some of the axioms are diﬃcult to verify and the description is far from a classiﬁcation.In this paper we present a very surprising fact which shows that towers of algebras giving rise to combinatorial Hopf algebras are much more rigid than they appear. Theorem1.1.If A= n≥0A n is a tower of algebras such that its associated Grothendieck groups form a pair of graded dual Hopf algebras,then dim(A n)=r n n! where r=dim(A1).The notion of“forming a pair of graded dual Hopf algebras”is made precise in Section2.The numbers r n n!will be familiar to experts in the theory of dual graded graphs–they count certain paths in a pair of dual graded graphs.The rigidity proved in Theorem1.1suggests that there may be a structure the-orem for towers of algebras which give rise to combinatorial Hopf algebras.In particular,to perform the inverse constructions of the arrows in(1.1),it suggests that one should study algebras related to symmetric groups(or wreath products of symmetric groups).There are many combinatorial Hopf algebras for which one may attempt to perform the inverse construction,but there are even more dual graded graphs.The general construction of[10]produces dual graded graphs from Bruhat orders of Weyl groups of Kac-Moody algebras and it is unclear whether there are Hopf algebras,or towers of algebras giving rise to these graphs.Acknowledgements.The second author would like to thank Mark Shimozono for the collaboration which led to the line of thinking in this paper.2.From towers of algebras to combinatorial Hopf algebrasWe recall here the work of Bergeron and Li[3]on towers of algebras.For B an arbitrary algebra we denote by B mod,the category of allﬁnitely generated left B-modules,and by P(B),the category of allﬁnitely generated projective left B-modules.For some category C of left B-modules(B mod or P(B))let F be the free abelian group generated by the symbols(M),one for each isomorphism class of modules M in C.Let F0be the subgroup of F generated by all expressions (M)−(L)−(N)one for each exact sequence0→L→M→N→0in C.The Grothendieck group K0(C)of the category C is deﬁned by the quotient F/F0,an abelian additive group.For M∈C,we denote by[M]its image in K0(C).COMBINATORIAL HOPF ALGEBRAS AND TOWERS OF ALGEBRAS3 We then setG0(B)=K0(B mod)and K0(B)=K0(P(B)).For B aﬁnite-dimensional algebra over aﬁeld K,let{V1,···,V s}be a complete list of nonisomorphic simple B-modules.The projective covers{P1,···,P s}of the simple modules V i’s is a complete list of nonisomorphic indecomposable projective B-modules.We have that G0(B)= s i=1Z[V i]and K0(B)= s i=1Z[P i].Letϕ:B→A be an injection of algebras preserving unities,and let M be a(left)A-module and N a(left)B-module.The induction of N from B to A is Ind A B N=A⊗ϕN,the(left)A-module A⊗N modulo the relations a⊗bn≡aϕ(b)⊗n, and the restriction of M from A to B is Res A B M=Hom A(A,M),the(left)B-module with the B-action deﬁned by bf(a)=f(aϕ(b)).Let A= n≥0A n be a graded algebra over C with multiplicationρ:A⊗A→A. Bergeron and Li studiedﬁve axioms for A(we refer to[3]for full details):(1)For each n≥0,A n is aﬁnite-dimensional algebra by itself with(internal) multiplicationµn:A n⊗A n→A n and unit1n.A0∼=C.(2)The(external)multiplicationρm,n:A m⊗A n→A m+n is an injective homo-morphism of algebras,for all m and n(sending1m⊗1n to1m+n).(3)A m+n is a two-sided projective A m⊗A n-module with the action deﬁned by a·(b⊗c)=aρm,n(b⊗c)and(b⊗c)·a=ρm,n(b⊗c)a,for all m,n≥0,a∈A m+n,b∈A m,c∈A n and m,n≥0.(4)A relation between the decomposition of A n+m as a left A m⊗A n-module and as a right A m⊗A n-module holds.(5)An analogue of Mackey’s formula relating induction and restriction of modules holds.We say here that A= n≥0A n is a tower of algebras if it satisﬁes Conditions (1),(2)and(3).Condition(1)guarantees that the Grothendieck groups G(A)= n≥0G0(A n) and K(A)= n≥0K0(A n)are graded connected.Conditions(2)and(3)ensure that induction and restriction are well deﬁned on G(A)and K(A),deﬁning a mul-tiplication and comultiplication,as follows.For[M]∈G0(A m)(or K0(A m))and [N]∈G0(A n)(or K0(A n))we let[M][N]= Ind A m+n A m⊗A n M⊗N and∆([N])= k+l=n Res A k+l A k⊗A l N . The pairing between K(A)and G(A)is given by , :K(A)×G(A)→Z where [P],[M] = dim K Hom A n(P,M) if[P]∈K0(A n)and[M]∈G0(A n),0otherwise.Thus with(only)Conditions(1),(2),and(3),G(A)and K(A)are dual free Z-modules both endowed with a multiplication and comultiplication.Bergeron and Li[3]proveTheorem2.1.If a graded algebra A= n≥0A n over C satisﬁes Conditions(1)-(5)then G(A)and K(A)are graded dual Hopf algebras.In particular Theorem1.1applies to graded algebras which satisfy Conditions (1)-(5).Note that the dual Hopf algebras G(A)and K(A)come with distinguished4NANTEL BERGERON,THOMAS LAM,AND HUILAN LIbases consisting of the isomorphism classes of simple and indecomposable projective modules.3.From combinatorial Hopf algebras to dual graded graphsThis section recounts work of Fomin[5],Hivert and Nzeutchap[8],and Lam and Shimozono.A graded graphΓ=(V,E,h,m)consists of a set of vertices V,a set of(directed)edges E⊂V×V,a height function h:V→{0,1,...}and an edge multiplicity function m:V×V→{0,1,...}.If(v,u)∈E is an edge then we must have h(u)=h(v)+1.The multiplicity function determines the edge set:(v,u)∈E if and only if m(v,u)=0.We assume always that there is a single vertex v0of height0.Let Z V= v∈V Z·v be the free Z-module generated by the vertex set.Given a graded graphsΓ=(V,E,h,m)we deﬁne up and down operators U,D:Z V→Z V byUΓ(v)= u∈V m(v,u)u DΓ(v)= u∈V m(u,v)uand extending by linearity over Z.We will assume thatΓis locally-ﬁnite,so that these operators are well deﬁned.A pair(Γ,Γ′)of graded graphs with the same vertex set V and height function h is called dual with diﬀerential coeﬃcient r if we haveDΓ′UΓ−UΓDΓ′=r Id.We shall need the following result of Fomin.For a graded graphΓ,let f vΓdenote the number of paths from v0to v,where for two vertices w,u∈V,we think that there are m(w,u)edges connecting w to u.Theorem3.1(Fomin[5]).Let(Γ,Γ′)be a pair of dual graded graphs with diﬀer-ential coeﬃcient r.Thenr n n!= v:h(v)=n f vΓf vΓ′.Let H•= n≥0H n and H•= n≥0H n be graded dual Hopf algebras over Z with respect to the pairing .,. :H•×H•→Z.We assume that we are given dual sets of homogeneous free Z-module generators{pλ∈H•}λ∈Λand{sλ∈H•}λ∈Λ, such that all structure constants are non-negative integers.We also assume that dim(H i)=dim(H i)<∞for each i≥0and dim(H0)=dim(H0)=1,so that H0 and H0are spanned by distinguished elements the unit1.Let us suppose we are given non-zero homogeneous elementsα∈H1andβ∈H1of degree1.We now deﬁne a graded graphΓ(β)=(V,E,h,m)where V={sλ}λ∈Λand h:V→Z is deﬁned by h(sλ)=deg(sλ).The map m:V×V→Z is deﬁned bym(sλ,sµ)= pµ,βsλ = ∆(pµ),β⊗sλand E is determined by m.The graphΓ(β)is graded because of the assumption thatβhas degree1.Similarly,we deﬁne a graded graphΓ′(α)=(V′,E′,h′,m′) where V′=V,h′=h,andm′(sλ,sµ)= αpλ,sµ = α⊗pλ,∆(sµ) .The following theorem is due independently to Hivert and Nzeutchap[8]and Lam and Shimozono(unpublished).COMBINATORIAL HOPF ALGEBRAS AND TOWERS OF ALGEBRAS 5Theorem 3.2.The graded graphs Γ=Γ(β)and Γ′=Γ′(α)form a pair of dual graded graphs with diﬀerential coeﬃcient α,β .Proof.We identify Z V with H •and note that U Γ(x )=βx where x ∈H •and we use the multiplication in H •.Also,D Γ′(x )= µ∈Λ α⊗p µ,∆x s µ= α,x (1) x (2).where ∆x = x (1)⊗x (2).Now observe that by our hypotheses on the degree of αand βthey are primitive elements:∆α=1⊗α+α⊗1and ∆β=1⊗β+β⊗1.We ﬁrst calculateα,βx = ∆α,β⊗x = 1,β α,x + α,β 1,x = α,β 1,xand then computeD Γ′U Γ(x )=D Γ′(βx )=α,βx (1) x (2)+ α,x (1) βx (2) = α,β x +U ΓD Γ′(x )where to obtain α,β x in the last line we use ∆x =1⊗x +terms of other degrees .4.Proof of Theorem 1.1We are given a graded algebra A = n ≥0A n over C with multiplication ρsatis-fying Conditions (1),(2)and (3).Moreover we assume that the two Grothendieck groups G (A )and K (A )form a pair of graded dual Hopf algebras as in Section 2.Under these assumptions we show thatdim(A n )=r n n !where r =dim(A 1).Let H •=G (A )and H •=K (A ).Let {s (1)1=[S (1)1],...,s (1)t =[S (1)t ]}and{p (1)1=[P (1)1],...,p (1)t =[P (1)t ]}denote the isomorphism classes of simple and inde-composable projective A 1-modules,so that H 1= t i =1Z s (1)i and H 1= t i =1Z p (1)i .Deﬁne a i =dim(S (1)i )and b i =dim(P (1)i )for 1≤i ≤t .We set for the remainder of this paperα=t i =1a i p (1)i∈H 1and β=t i =1b i s (1)i ∈H 1.Since A 0∼=C ,we let s (0)1(respectively,p (0)1)be the unique simple (respectively,indecomposable projective)module representative in H 0(respectively,H 0).Simi-larly,let {s (n )i =[S (n )i ]}be all isomorphic classes of simple A n -modules and {p (n )i =[P (n )i ]}be all isomorphism classes of indecomposable projective A n -modules.The sets n ≥0{s (n )i }and n ≥0{p (n )i }form dual free Z -module bases of H •and H •.Now deﬁne Γ=Γ(β)and Γ′=Γ′(α)as in Section 3.Lemma 4.1.We havef s (n )j Γ=dim P (n )j and f s (n )jΓ′=dim S (n )j .6NANTEL BERGERON,THOMAS LAM,AND HUILAN LI Proof.We havem(s(n−1)i ,s(n)j)=tl=1b l c l,where c l is the number of copies of the indecomposable projective module P(1)l⊗P(n−1) i as a summand in Res A n A1⊗A n−1P(n)j.Note that s(0)1is the unit of H•andm(s(0)1,s(1)i)=b i=dim P(1)ifor all1≤i≤t.The dimension of an indecomposableprojective module P(n)jis given bydim P(n)j= i,l c l dim P(1)l⊗P(n−1)i = i m(s(n−1)i,s(n)j)dim P(n−1)i.By induction on n,we deduce that dim P(n)jis the number of paths from s(0)1tos(n)jinΓ.The claim forΓ′is similar.For anyﬁnite dimensional algebra B let{Sλ}λbe a complete set of simple B-modules.For eachλlet Pλbe the projective cover of Sλ.It is well known(see[4]) that we canﬁnd minimal idempotents{e i}such that B= Be i where each Be i is isomorhpic to a Pλ.Moreover,the quotient of B by its radical shows that the multiplicity of Pλin B is equal to dim Sλ.This implies the following lemma. Lemma4.2.Let B be aﬁnite dimensional algebra and{Sλ}λbe a complete set of simple B-modules.dim B= λ(dim Pλ)(dim Sλ),where Pλis the projective cover of Sλ.By Lemma4.2,we have r= t i=1a i b i= α,β .By Theorem3.2we may apply Theorem3.1to(Γ,Γ′).Using Lemma4.2and Lemma4.1,Theorem3.1says dim(A n)= i(dim P(n)i)(dim S(n)i)= i f s(n)iΓf s(n)iΓ′=r n n!.Remark4.3.If the tower consists of semisimple algebras A i thenΓ=Γ′so we obtain a self-dual graph.In this case the graph would be a weighted version of a diﬀerential poset in the sense of Stanley[13].If furthermore the branching of irreducible modules from A n to A1⊗A n−1is multiplicity free then we get a true diﬀerential poset.Remark4.4.The Hopf algebras H•and H•are not in general commutative and co-commutative.Thus in the deﬁnitions of Section3we could have obtained a diﬀerent pair of dual graded graphs by setting m(sλ,sµ)= pµ,sλβ or m′(sλ,sµ)= pλα,sµ .References[1]M.Aguiar,N.Bergeron,and F.Sottile,Combinatorial Hopf algebras and generalized Dehn-Sommerville relations,Compos.Math.142(2006),no.1,1–30.[2]N.Bergeron,F.Hivert and J.Y.Thibon,The peak algebra and the Hecke-Cliﬀord algebrasat q=bin.Theory Ser.A107-1(2004)1–19.[3]N.Bergeron and H.Li,Algebraic Structures on Grothendieck Groups of a Tower of Algebras,To appear.[arXiv:math/0612170].COMBINATORIAL HOPF ALGEBRAS AND TOWERS OF ALGEBRAS7 [4]C.Curtis and I.Reiner,Methods of representation theory.Vol.I.With applications toﬁnitegroups and orders,John Wiley&Sons,Inc.,New York,1990.[5]S.Fomin,Duality of graded graphs,J.Algebraic Combin.3(1994),no.4,357–404.[6]L.Geissinger,Hopf algebras of symmetric functions and class functions,Combinatoire etCombinatoire et repr´e sentation du groupe sym´e trique(Actes Table Ronde C.N.R.S.,Univ.Louis-Pasteur Strasbourg,Strasbourg,1976),pp.168–181.Lecture Notes in Math.,Vol.579, Springer,Berlin,1977.[7]F.Hivert,J.-C.Novelli and J.-Y.Thibon,Representation theory of the0-Ariki-Koike-Shojialgebras,to appear.[math.CO/0407218].[8]F.Hivert and eutchap,Dual Graded Graphs in Combinatorial Hopf Algebras,in prepa-ration.[9]D.Krob and J.Y.Thibon,Noncommutative symmetric functions.IV.Quantum linear groupsand Hecke algebras at q=0,J.Algebraic Combin.6-4(1997)339–376.[10]m and M.Shimozono,Dual graded graphs for Kac-Moody algebras,Algebra and NumberTheory,to appear.[math.CO/0702090].[11]C.Malvenuto and C.Reutenauer,Duality between quasi-symmetric functions and thesolomon descent algebra,J.Algebra177-3(1995)967–982.[12]A.N.Sergeev,Tensor algebra of the identity representation as a module over the Lie super-algebras GL(n;m)and Q(n),SR Sbornik51(1985)419–427.[13]R.Stanley,Diﬀerential posets,J.Amer.Math.Soc.1(1988),919–961.[14]A.V.Zelevinsky,Representations ofﬁnite classical groups.A Hopf algebra approach,LectureNotes in Mathematics869.Springer-Verlag,Berlin-New York,1981.(Nantel Bergeron)Department of Mathematics and Statistics,York University,To-ronto,Ontario M3J1P3,CANADAE-mail address:bergeron@mathstat.yorku.caURL:http://www.math.yorku.ca/bergeron(Thomas Lam)Department of Mathematics,Harvard University,Cambridge,,MA 02138.E-mail address:tfylam@URL:/~tfylam(Huilan Li)Department of Mathematics and Statistics,York University,Toronto, Ontario M3J1P3,CANADAE-mail address:lihuilan@mathstat.yorku.ca。

Exponential map - Wikipedia, the free encyclopedia

The exponential map of the Earth as viewed from the north pole is the polar azimuthal equidistant projection in cartography.Exponential mapFrom Wikipedia, the free encyclopediaIn differential geometry, the exponential map is ageneralization of the ordinary exponential functionof mathematical analysis to all differentiablemanifolds with an affine connection. Two importantspecial cases of this are the exponential map for amanifold with a Riemannian metric, and theexponential map from a Lie algebra to a Lie group.Contents1 Definition2 Lie theory2.1 Definitions2.2 Examples2.3 Properties3 Riemannian geometry3.1 Properties4 Relationships5 See also6 Notes7 References DefinitionLet M be a differentiable manifold and p a point of M . An affine connection on M allows one to define the notion of a geodesic through the point p .[1]Let v ∈ T p M be a tangent vector to the manifold at p . Then there is a uniquegeodesic γv satisfying γv (0) = p with initial tangent vector γ′v (0) = v . The corresponding exponential map is defined by exp p (v ) = γv (1). In general, theexponential map is only locally defined , that is, it only takes a small neighborhood of the origin at T p M , to a neighborhood of p in the manifold. This is because it relies on the theorem on existence and uniqueness for ordinary differentialequations which is local in nature. An affine connection is called complete if the exponential map is well-defined at every point of the tangent bundle.Lie theoryIn the theory of Lie groups, the exponential map is a map from the Lie algebra ofa Lie group to the group which allows one to recapture the local group structure from the Lie algebra. The existence of the exponential map is one of the primary justifications for the study of Lie groups at the level of Lie algebras.The ordinary exponential function of mathematical analysis is a special case of the exponential map when G is the multiplicative group of positive real numbers (whose Lie algebra is the additive group of all real numbers). The exponential map of a Lie group satisfies many properties analogous to those of the ordinary exponential function, however, it also differs in many important respects.DefinitionsLet be a Lie group and be its Lie algebra (thought of as the tangent space to the identity element of ). The exponential map is a mapwhich can be defined in several different ways as follows:It is the exponential map of a canonical left-invariant affine connection on G, such that parallel transport is given by left translation.It is the exponential map of a canonical right-invariant affine connection onG. This is usually different from the canonical left-invariant connection, butboth connections have the same geodesics (orbits of 1-parameter subgroupsacting by left or right multiplication) so give the same exponential map.It is given by whereis the unique one-parameter subgroup of whose tangent vector at the identity is equal to . It follows easily from the chain rule that .The map may be constructed as the integral curve of either the right- orleft-invariant vector field associated with . That the integral curve exists for all real parameters follows by right- or left-translating the solution near zero.If is a matrix Lie group, then the exponential map coincides with the matrix exponential and is given by the ordinary series expansion:(here is the identity matrix).If G is compact, it has a Riemannian metric invariant under left and righttranslations, and the exponential map is the exponential map of this Riemannianmetric.ExamplesThe unit circle centered at 0 in the complex plane is a Lie group (called the circle group) whose tangent space at 1 can be identified with the imaginary line in the complex plane, The exponential map for this Lie group is given bythat is, the same formula as the ordinary complex exponential.In the split-complex number plane the imaginary line forms the Lie algebra of the unit hyperbola groupsince the exponential map is given byThe unit 3-sphere centered at 0 in the quaternions H is a Lie group(isomorphic to the special unitary group ) whose tangent space at 1 can be identified with the space of purely imaginary quaternions,The exponential map for this Lie group is given byThis map takes the 2-sphere of radius inside the purely imaginaryquaternions to a 2-sphere of radiuswhen . Compare this to the first example above.PropertiesFor all , the map is the unique one-parameter subgroup of whose tangent vector at the identity is . It follows that:The exponential map is a smooth map. Its derivative at theidentity, , is the identity map (with the usual identifications).The exponential map, therefore, restricts to a diffeomorphism from someneighborhood of 0 in to a neighborhood of 1 in .The exponential map is not, however, a covering map in general – it is not alocal diffeomorphism at all points. For example, so(3) to SO(3) is not acovering map; see also cut locus on this failure.The image of the exponential map always lies in the identity component of .When is compact, the exponential map is surjective onto the identitycomponent.The image of the exponential map of the connected but non-compact group SL2(R) is not the whole group. Its image consists of C-diagonalizable matrices with eigenvalues either positive or with module 1, and of non-diagonalizabletrigonalizable matrices with eigenvalue 1.The map is the integral curve through the identity of both the right- and left-invariant vector fields associated to .The integral curve through of the left-invariant vector fieldassociated to is given by . Likewise, the integral curve through of the right-invariant vector field is given by . It follows that the flows generated by the vector fields are given by:Since these flows are globally defined, every left- and right-invariant vector field on is complete.Let be a Lie group homomorphism and let be its derivative at theidentity. Then the following diagram commutes:In particular, when applied to the adjoint action of a group we haveRiemannian geometryIn Riemannian geometry, an exponential map is a map from a subset of a tangent space T p M of a Riemannian manifold (or pseudo-Riemannian manifold) M to M itself. The (pseudo) Riemannian metric determines a canonical affine connection, and the exponential map of the (pseudo) Riemannian manifold is given by the exponential map of this connection.PropertiesIntuitively speaking, the exponential map takes a given tangent vector to the manifold, runs along the geodesic starting at that point and going in that direction, for a unit time. Since v corresponds to the velocity vector of the geodesic, the actual (Riemannian) distance traveled will be dependent on that. We can also reparametrize geodesics to be unit speed, so equivalently we can defineexp p(v) = β(|v|) where β is the unit-speed geodesic (geodesic parameterized by arc length) going in the direction of v. As we vary the tangent vector v we will get, when applying exp p, different points on M which are within some distance from the base point p—this is perhaps one of the most concrete ways of demonstrating that the tangent space to a manifold is a kind of "linearization" of the manifold.The Hopf–Rinow theorem asserts that it is possible to define the exponential map on the whole tangent space if and only if the manifold is complete as a metric space (which justifies the usual term geodesically complete for a manifold having an exponential map with this property). In particular, compact manifolds are geodesically complete. However even if exp p is defined on the whole tangent space,it will in general not be a global diffeomorphism. However, its differential at the origin of the tangent space is the identity map and so, by the inverse function theorem we can find a neighborhood of the origin of T p M on which the exponential map is an embedding (i.e., the exponential map is a local diffeomorphism). The radius of the largest ball about the origin in T p M that can be mapped diffeomorphically via exp p is called the injectivity radius of M at p. The cut locus of the exponential map is, roughly speaking, the set of all points where the exponential map fails to have a unique minimum.An important property of the exponential map is the following lemma of Gauss (yet another Gauss's lemma): given any tangent vector v in the domain of definition of exp p, and another vector w based at the tip of v (hence w is actually in the double-tangent space T v(T p M)) and orthogonal to v, remains orthogonal to v when pushed forward via the exponential map. This means, in particular, that the boundary sphere of a small ball about the origin in T p M is orthogonal to the geodesics in M determined by those vectors (i.e., the geodesics are radial). This motivates the definition of geodesic normal coordinates on a Riemannian manifold.The exponential map is also useful in relating the abstract definition of curvature to the more concrete realization of it originally conceived by Riemann himself—the sectional curvature is intuitively defined as the Gaussian curvature of some surface (i.e., a slicing of the manifold by a 2-dimensional submanifold) through the point p in consideration. Via the exponential map, it now can be precisely defined as the Gaussian curvature of a surface through p determined by the image under exp p of a 2-dimensional subspace of T p M.RelationshipsIn the case of Lie groups with a bi-invariant metric—a pseudo-Riemannian metric invariant under both left and right translation—the exponential maps of the pseudo-Riemannian structure are the same as the exponential maps of the Lie group. In general, Lie groups do not have a bi-invariant metric, though all connected semi-simple (or reductive) Lie groups do. The existence of a bi-invariant Riemannian metric is stronger than that of a pseudo-Riemannian metric, and implies that the Lie algebra is the Lie algebra of a compact Lie group; conversely, any compact (or abelian) Lie group has such a Riemannian metric.Take the example that gives the "honest" exponential map. Consider the positive real numbers R+, a Lie group under the usual multiplication. Then each tangent space is just R. On each copy of R at the point y, we introduce the modified inner product(multiplying them as usual real numbers but scaling by y2). (This is what makes the metric left-invariant, for left multiplication by a factor will just pull out of the inner product, twice — canceling the square in the denominator).Consider the point 1 ∈ R+, and x ∈ R an element of the tangent space at 1. The usual straight line emanating from 1, namely y(t) = 1 + xt covers the same path as a geodesic, of course, except we have to reparametrize so as to get a curve with constant speed ("constant speed", remember, is not going to be the ordinary constant speed, because we're using this funny metric). To do this we reparametrize by arc length (the integral of the length of the tangent vector in the norm induced by the modified metric):and after inverting the function to obtain t as a function of s, we substitute and getNow using the unit speed definition, we have,giving the expected e x.The Riemannian distance defined by this is simply,a metric which should be familiar to anyone who has drawn graphs on log paper. See alsoList of exponential topicsNotes1. ^ A source for this section is Kobayashi & Nomizu (1975, §III.6), which uses the term"linear connection" where we use "affine connection" instead.Referencesdo Carmo, Manfredo P. (1992), Riemannian Geometry, Birkhäuser, ISBN 0-8176-3490-8. See Chapter 3.Cheeger, Jeff; Ebin, David G. (1975), Comparison Theorems in RiemannianGeometry, Elsevier. See Chapter 1, Sections 2 and 3.Hazewinkel, Michiel, ed. (2001), "Exponential mapping"(/index.php?title=p/e036930), Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4Helgason, Sigurdur (2001), Differential geometry, Lie groups, and symmetric spaces, Graduate Studies in Mathematics 34, Providence, R.I.: AmericanMathematical Society, ISBN 978-0-8218-2848-9, MR 1834454(https:///mathscinet-getitem?mr=1834454).Kobayashi, Shoshichi; Nomizu, Katsumi (1996), Foundations of DifferentialGeometry, Vol. 1 (New ed.), Wiley-Interscience, ISBN 0-471-15733-3.Retrieved from "/w/index.php?title=Exponential_map&oldid=602570281"Categories: Exponentials Lie groups Riemannian geometryThis page was last modified on 3 April 2014 at 12:36.Text is available under the Creative Commons Attribution-ShareAlike License;additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia® is a registered trademark of the WikimediaFoundation, Inc., a non-profit organization.。

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of particles in plasmas, 5o2f .25.Tx photons by nuclei, 25.20.Dc
Absorpti on coefficie nts, optical, 78.20.Ci
Absorpti on edges, xray, 78.70.D m
Absorpti on spectra atoms, 32.30.-r
architect ural acoustic s, *43.55.E v, *43.55.N d
linear acoustic s, *43.20.H q
nonlinear acoustic s, *43.25.E d
underwat er, *43.30.E s, *43.30.K y
by atoms, 32.80.-t
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A15 compoun ds and alloys, 74.70.Ad
Aberratio ns, optical, 42.15.Fr
Ab initio calculati ons (atoms and molecule s), 31.15.AAblation
Acoustoelectric effects
conducti vity phenom ena, 72.50.+b
piezoele ctrics, 7th7in.65.Dq films, 73.50.Rb
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Automatic mapping of ASSIST applications using process algebraMarco AldinucciDept.of Computer Science,University of PisaLargo B.Pontecorvo3,Pisa I-56127,ItalyandAnne BenoitLIP,Ecole Normale Superieure de Lyon(ENS)46all´e e d’Italie,69364Lyon Cedex07,FranceReceived(received date)Revised(revised date)Communicated by(Name of Editor)ABSTRACTGrid technologies aim to harness the computational capabilities of widely distributed collections of computers.Due to the heterogeneous and dynamic nature of the set of grid resources,the programming and optimisation burden of a low level approach to grid computing is clearly unacceptable for large scale,complex applications.The development of grid applications can be simpliﬁed by using high-level programming environments.In the present work,we address the problem of the mapping of a high-level grid application onto the computational resources.In order to optimise the mapping of the application, we propose to automatically generate performance models from the application using the process algebra PEPA.We target applications written with the high-level environment ASSIST,since the use of such a structured environment allows us to automate the study of the application more eﬀectively.Keywords:high-level parallel programming;ASSIST environment;Performance Eval-uation Process Algebra(PEPA);automatic model generation.1.IntroductionA grid system is a geographically distributed collection of possibly parallel,inter-connected processing elements,which all run some form of common grid middleware (e.g.Globus)[13].The key idea behind grid-aware applications is to make use of the aggregate power of distributed resources,thus beneﬁting from a computing power that falls far beyond the current availability threshold in a single site.However, developing programs able to exploit this potential is highly programming inten-sive.Programmers must design concurrent programs that can execute on large-scale platforms that cannot be assumed to be homogeneous,secure,reliable or centrally managed.They must then implement these programs correctly and eﬃciently.As a result,in order to build eﬃcient grid-aware applications,programmers have to address the classical problems of parallel computing as well as grid-speciﬁc ones:Parallel Processing Letters1.Programming:code all the program details,take care about concurrency ex-ploitation,among the others:concurrent activities set up,mapping/scheduling, communication/synchronisation handling and data allocation.2.Mapping&Deploying:deploy application processes according to a suitablemapping onto grid platforms.These may be highly heterogeneous in archi-tecture and performance and unevenly connected,thus exhibiting diﬀerent connectivity properties among all pairs of platforms.3.Dynamic environment:manage resource unreliability and dynamic availabil-ity,network topology,latency and bandwidth unsteadiness.Hence,the number and quality of problems to be resolved in order to draw a given QoS(in term of performance,robustness,etc.)from grid-aware applications is quite large.The lesson learnt from parallel computing suggests that any low-level approach to grid programming is likely to raise the programmer’s burden to an unacceptable level for any real world application.Therefore,we envision a layered, high-level programming model for the grid,which is currently pursued by several research initiatives and programming environments,such as ASSIST[19],eSkel[9], GrADS[17],ProActive[6],Ibis[18].In such an environment,most of the grid speciﬁc eﬀorts are moved from programmers to grid tools and run-time systems. Thus,the programmers have only the responsibility of organising the application speciﬁc code,while the developing tools and their run-time systems deal with the interaction with the grid,through collective protocols and services[12].In such a scenario,the QoS and performance constraints of the application can either be speciﬁed at compile time or varying at run-time.In both cases,the run-time system should actively operate in order to fulﬁl QoS requirements of the application,since any static resource assignment may violate QoS constraints due to the very uneven performance of grid resources over time.As an example,AS-SIST applications exploit an autonomic(self-optimisation)behaviour.They may be equipped with a QoS contract describing the degree of performance the application is required to provide.The ASSIST run-time environment tries to keep the QoS contract valid for the duration of the application run despite possible variations of platforms’performance at the level of grid fabric[5].The autonomic features of an ASSIST application rely heavily on run-time application monitoring,and thus they are not fully eﬀective for application deployment since the application is not yet running.In order to deploy an application onto the grid,a suitable mapping of application processes onto grid platforms should be established,and this process is quite critical for application performance.This problem can be addressed by deﬁning a performance model of an ASSIST application in order to statically optimise the mapping of the application onto a heterogeneous environment.The model is generated from the source code of the application,before the initial mapping.It is expressed with the process algebra PEPA[15],designed for performance evaluation.The use of a stochastic model allows us to take into account aspects of uncertainty which are inherent to grid computing,and to use classical techniques of resolution based on Markov chains toAutomatic mapping of ASSIST applications using process algebra obtain performance results.This static analysis of the application is complemen-tary with the autonomic reconﬁguration of ASSIST applications,which works on a dynamic basis.In this work we concentrate on the static part to optimise the map-ping,while the dynamic management is done at run-time.It is thus an orthogonal but complementary approach.Structure of the paper.The next section introduces the ASSIST high-level pro-gramming environment and its run-time support.Section3introduces the Per-formance Evaluation Process Algebra PEPA,which can be used to model ASSIST applications.These performance models help to optimise the mapping of the ap-plication.We present our approach in Section4,and give an overview of future working directions.Finally,concluding remarks are given in Section5.2.The ASSIST environment and its run-time supportASSIST(A Software System based on Integrated Skeleton Technology)is a pro-gramming environment aimed at the development of distributed high-performance applications[19,3].ASSIST applications should be compiled in binary packages that can be deployed and run on grids,including those exhibiting heterogeneous platforms.Deployment and run is provided through standard middleware services(e.g.Globus)enriched with the ASSIST run-time support.2.1.The ASSIST coordination languageASSIST applications are described by means of a coordination language,which can express arbitrary graphs of modules,interconnected by typed streams of data. Each stream realises a one-way asynchronous channel between two sets of endpoint modules:sources and sinks.Data items injected from sources are broadcast to all sinks.Modules can be either sequential or parallel.A sequential module wraps a sequential function.A parallel module(parmod)can be used to describe the parallel execution of a number of sequential functions that are activated and run as Virtual Processes(VPs)on items arriving from input streams.The VPs may synchronise with the others through barriers.The sequential functions can be programmed by using a standard sequential language(C,C++,Fortran,Java).A parmod may behave in a data-parallel(e.g.SPMD/apply-to-all)or task-parallel(e.g.farm)way and it may exploit a distributed shared state that survives the VPs lifespan.A module can nondeterministically accept from one or more input streams a number of input items,which may be decomposed in parts and used as function parameters to instantiate VPs according to the input and distribution rules speciﬁed in the parmod.The VPs may send items or parts of items onto the output streams,and these are gathered according to the output rules.An ASSIST application is sketched in Appendix A.We brieﬂy describe here how to code an ASSIST application and its modules;more details on the particular application in Appendix A are given in Section4.1.In lines4–5four streams with type task t are declared.Lines6–9deﬁne endpoints of streams.Overall,Parallel Processing Letters ASSISTcompiler seq P1parmod VP VP VP binary les QoScontract ASSIST programresourcedescription(XML)VP VP VP VP VP VP VP VP VP output section input section binary code+XML (network of processes)ISM OSM P1P2VP VPVP VP VP VPMVP seq P2sourcecode Fig.An ASSIST application QoS contract are compiled in a set of executable codes and its meta-data [3].This information is used to set up a processes network at launch time.lines 3–10deﬁne the application graph of modules.In lines 12–16two sequential modules are declared:these simply provide a container for a sequential function invocation and the binding between streams and function parameters.In lines 18–52two parmods are declared.Each parmod is characterised by its topology ,input section ,virtual processes ,and output section declarations.The topology declaration specialises the behaviour of the VPs as farm (topol-ogy none ,as in line 41),or SMPD (topology array ).The input section en-ables programmers to declare how VPs receive data items,or parts of items,from streams.A single data item may be distributed (scattered,broadcast or unicast)to many VPs.The input section realises a CSP repetitive command [16].The virtual processes declarations enable the programmer to realise a parametric VP starting from a sequential function (proc ).VPs may be identiﬁed by an index and may synchronise and exchange data one with another through the ASSIST lan-guage API.The output section enables programmers to declare how data should be gathered from VPs to be sent onto output streams.More details on the ASSIST coordination language can be found in [19,3,2].2.2.The ASSIST run-time supportThe ASSIST compiler translates a graph of modules into a network of processes.As sketched in Fig.1,sequential modules are translated into sequential processes,while parallel modules are translated into a parametric (w.r.t.the parallelism de-gree)network of processes:one Input Section Manager (ISM),one Output Section Manager (OSM),and a set of Virtual Processes Managers (VPMs,each of them running a set of Virtual Processes).The actual parallelism degree of a parmod instance is given by the number of VPMs.All processes communicate via ASSIST support channels,which can be implemented on top of a number of grid middleware communication mechanisms (e.g.shared memory,TCP/IP,Globus,CORBA-IIOP,SOAP-WS).The suitable communication mechanism between each pair of processes is selected at launch time depending on the mapping of the processes [3].Automatic mapping of ASSIST applications using process algebra 2.3.Towards fully grid-aware applicationsASSIST applications can already cope with platform heterogeneity,either in space(various architectures)or in time(varying load)[5,2].These are deﬁnite fea-tures of a grid,however they are not the only ones.Grids are usually organised in sites on which processing elements are organised in networks with private ad-dresses allowing only outbound connections.Also,they are often fed through job schedulers.In these cases,setting up a multi-site parallel application onto the grid is a challenge in its own right(irrespectively of its performance).Advance reser-vation,co-allocation,multi-site launching are currently hot topics of research for a large part of the grid community.Nevertheless,many of these problems should be targeted at the middleware layer level and they are largely independent of the logical mapping of application processes on a suitable set of resources,given that the mapping is consistent with deployment constraints.In our work,we assume that the middleware level supplies(or will supply) suitable services for co-allocation,staging and execution.These are actually the minimal requirements in order to imagine the bare existence of any non-trivial, multi-site parallel application.Thus we can analyse how to map an ASSIST ap-plication,assuming that we can exploit middleware tools to deploy and launch applications[11].3.Introduction to performance evaluation and PEPAIn this section,we brieﬂy introduce the Performance Evaluation Process Algebra PEPA[15],with which we can model an ASSIST application.The use of a process algebra allows us to include the aspects of uncertainty relative to both the grid and the application,and to use standard methods to easily and quickly obtain performance results.The PEPA language provides a small set of combinators. These allow language terms to be constructed deﬁning the behaviour of components, via the activities they undertake and the interactions between them.We can for instance deﬁne constants(def=),express the sequential behavior of a given component (.),a choice between diﬀerent behaviors(+),and the direct interaction betweencomponents(£¡L ,||).Timing information is associated with each activity.Thus,when enabled,an activity a=(α,r)will delay for a period sampled from the negative exponential distribution which has parameter r.If several activities are enabled concurrently,either in competition or independently,we assume that a race condition exists between them.When an activity is known to be carried out in cooperation with another component,a component may be passive with respect to that activity.This means that the rate of the activity is left unspeciﬁed,(denoted ),and is determined upon cooperation by the rate of the activity in the other component.All passive actions must be synchronised in theﬁnal model.The dynamic behaviour of a PEPA model is represented by the evolution of its components,as governed by the operational semantics of PEPA terms[15].Thus, as in classical process algebra,the semantics of each term is given via a labelledParallel Processing Lettersmulti-transition system(the multiplicity of arcs are signiﬁcant).In the transition system a state corresponds to each syntactic term of the language,or derivative,and an arc represents the activity which causes one derivative to evolve into another. The complete set of reachable states is termed the derivative set and these form the nodes of the derivation graph,which is formed by applying the semantic rules exhaustively.The derivation graph is the basis of the underlying Continuous Time Markov Chain(CTMC)which is used to derive performance measures from a PEPA model.The graph is systematically reduced to a form where it can be treated as the state transition diagram of the underlying CTMC.Each derivative is then a state in the CTMC.The transition rate between two derivatives P and Q in the derivation graph is the rate at which the system changes from behaving as component P to behaving as Q.Examples of derivation graphs can be found in[15].It is important to note that in our models the rates are represented as ran-dom variables,not constant values.These random variables are exponentially dis-tributed.Repeated samples from the distribution will follow the distribution and conform to the mean but individual samples may potentially take any positive value.The use of such distribution is quite realistic and it allows us to use stan-dard methods on CTMCs to readily obtain performance results.There are indeed several methods and tools available for analysing PEPA models.Thus,the PEPA Workbench[14]allows us to generate the state space of a PEPA model and the in-ﬁnitesimal generator matrix of the underlying Markov chain.The state space of the model is represented as a sparse matrix.The PEPA Workbench can then compute the steady-state probability distribution of the system,and performance measures such as throughput and utilisation can be directly computed from this.4.Performance models of ASSIST applicationPEPA can easily be used to model an ASSIST application since such applications are based on stream communications,and the graph structure deduced from these streams can be modelled with PEPA.Given the probabilistic information about the performance of each of the ASSIST modules and streams,we then aim toﬁnd information about the global behavior of the application,which is expressed by the steady-state of the system.The model thus allows us to predict the run-time behavior of the application in the long time run,taking into account information obtained from a static analysis of the program.This behavior is not known in advance,it is a result of the PEPA model.4.1.The ASSIST applicationAs we have seen in Section2,an ASSIST application consists of a series of modules and streams connecting the modules.The structure of the application is represented by a graph,where the modules are the nodes and the streams the arcs.We illustrate in this paper our modeling process on an example of a graph, but the process can be easily generalised to any ASSIST applications since theAutomatic mapping of ASSIST applications using process algebra M3M4M2M1s1s3s2Figure 2:Graph representation of our example application.information about the graph can be extracted directly from ASSIST source code,and the model can be generated automatically from the graph.A model of a data mining classiﬁcation algorithm has been presented in [1].For the purpose of our methodology and in order to generalise our approach,we concentrate here only on the graph of an application.The graph of the application that we consider in this paper is similar to the one of [1],consisting of four modules.Figure 2represents the graph of this application.We choose this graph as an application example,since this is a very common workﬂow pattern.In such a schema,•one module (M1)is generating input,for instance reading from a ﬁle or ac-cessing a database;•two modules (M2,M3)are interacting in a client-server way;they can interact one or several times for each input,in order to produce a result;•the result is sent to a last module (M4)which is in charge of the output.4.2.The PEPA modelEach ASSIST module is represented as a PEPA component,and the diﬀerent components are synchronised through the streams of data to model the overall application.The PEPA components of the modules are shown in Table 1.The modules are working in a sequential way:the module MX (X =1..4)is initially in the state MX1,waiting for data on its input streams.Then,in the state MX2,it processes the piece of data and evolves to its third state MX3.Finally,the module sends the output data on its output streams and goes back into its ﬁrst state.The system evolves from one state to another when an activity occurs.The activity sX (X =1..4)represents the transfer of data through the stream X ,with the associated rate λX .The rate reﬂects the complexity of the communication.The activity pX (X =1..4)represents the processing of a data by module MX ,which is done at a rate µX .These rates are related to the theoretical complexity of the modules.A discussion on rates is done in Section 4.3.The overall PEPA model is then obtained by a collaboration of the diﬀerent modules in their initial states:M 11£¡s 1M 21£¡s 2,s 3M 31£¡s 4M 41.4.3.Automatic generation of the modelThe PEPA model is automatically generated from the ASSIST source code.This task is simpliﬁed thanks to some information provided by the user directly in theParallel Processing Letterssource code,and particularly the rates associated to the diﬀerent activities of the PEPA model.The rates are directly related to the theoretical complexity of the modules and of the communications.In particular,rates of the communications depend on:a) the speed of the links and b)data size and communications frequencies.A module may include a parallel computation,thus its rate depends on a)computing power of the platforms running the module and b)parallel computation complexity,its size, its parallel degree,and its speedup.Observe that aspect a)of both modules and communications rates strictly depends on mapping,while aspect b)is much more dependent on the application’s logical structure and algorithms.We are interested in the relative computational and communication costs of the diﬀerent parts of the system,but we deﬁne numerical values to allow a numerical resolution of the PEPA model.This information is deﬁned directly in the ASSIST source code of the application by calling a rate function,in the body of the main procedure of the application(Appendix A,between lines9and10).This function takes as a parameter the name of the modules and streams,and it should be called once for each module and each stream toﬁx the rates of the corresponding PEPA activities.We can deﬁne several sets of rates in order to compare several PEPA models.The values for each sets are deﬁned between brackets,separated with commas,as shown in the example below.rate(s1)=(10,1000);rate(s2)=(10,1);rate(s3)=(10,1);rate(s4)=(10,1000);rate(M1)=(100,100);rate(M2)=(100,100);rate(M3)=(1,1);rate(M4)=(100,100);The PEPA model is generated during a precompilation of the source code of AS-SIST.The parser identiﬁes the main procedure and extracts the useful information from it:the modules and streams,the connections between them,and the rates of the diﬀerent activities.The main diﬃculty consists in identifying the schemes of input and output behaviour in the case of several streams.This information can be found in the input and output section of the parmod code.Regarding the input section,the parser looks at the guards.Details on the diﬀerent types of guards can be found in[19,3].Table1:PEPA model for the exampleM11def=M12M12def=(p1,µ1).M13M13def=(s1,λ1).M11M21def=(s1, ).M22+(s2, ).M22 M22def=(p2,µ2).M23M23def=(s3,λ3).M21+(s4,λ4).M21M31def=(s3, ).M32 M32def=(p3,µ3).M33 M33def=(s2,λ2).M31M41def=(s4, ).M42 M42def=(p4,µ4).M43 M43def=M41Automatic mapping of ASSIST applications using process algebra As an example,a disjoint guards means that the module takes input from ei-ther of the streams when some data arrives.This is translated by a choice in the PEPA model,as illustrated in our example.However,some more complex behaviour may also be expressed,for instance the parmod can be instructed to start execut-ing only when it has data from both streams.In this case,the PEPA model is changed with some sequential composition to express this behaviour.For example, M21def=(s1, ).(s2, ).M22+(s2, ).(s1, ).M22.Currently,we are not support-ing variables in guards,since these may change the frequency of accessing data on a stream.Since the variables may depend on the input data,we cannot automatically extract static information from them.We plan to address this problem by asking the programmer to provide the relative frequency of the guard.The considerations for the output section are similar.The PEPA model generated by the application for a given set of rates is repre-sented below:mu1=100;mu2=100;mu3=1;mu4=100;la1=10;la2=10;la3=10;la4=10;M11=M12;M12=(p1,mu1).M13;M13=(s1,la1).M11;M21=(s1,infty).M22+(s2,infty).M22;M22=(p2,mu2).M23;M23=(s3,la3).M21+(s4,infty).M21;M31=(s3,infty).M32;M32=(p3,mu3).M33;M33=(s2,la2).M31;M41=(s4,la4).M42;M42=(p4,mu4).M43;M43=M41;(M11<s1>(M21<s2,s3>M31))<s4>M414.4.Performance resultsOnce the PEPA models have been generated,performance results can be ob-tained easily with the PEPA Workbench[14].The performance results are the probability to be in either of the states of the system.We compute the probability to be waiting for a processing activity pX,or to wait for a transfer activity sX.Some additional information is generated in the PEPA source code(ﬁle example.pepa) to specify the performance results that we are interested in.This information is the following:perf_M1=100*{M12||**||**||**};perf_M2=100*{**||M22||**||**}; perf_M3=100*{**||**||M32||**};perf_M4=100*{**||**||**||M42}; perf_s1=100*{M13||M21||**||**};perf_s2=100*{**||M21||M33||**}; perf_s3=100*{**||M23||M31||**};perf_s4=100*{**||M23||**||M41};The expression in brackets describes the states of the PEPA model corresponding to a particular state of the system.For each module MX(X=1..4),the result perf MX corresponds to the percentage of time spent waiting to process this module.The steady-state probability is multiplied by100for readability and interpretation rea-sons.A similar result is obtained for each stream.We expect the complexity of the PEPA model to be quite simple and the resolution straightforward for most of the ASSIST applications.In our example,the PEPA model consists in36states andParallel Processing Letters80transitions,and it requires less than0.1seconds to generate the state space of the model and to compute the steady state solution,using the linear biconjugate gradient method[14].Experiment1.For the purpose of our example,we choose the following rates, meaning that the module M3is computationally more intensive than the other modules.In our case,M3has an average duration pared to0.01sec. for the others(µ1=100;µ2=100;µ3=1;µ4=100).The rates for the streams correspond to an average duration of0.1sec(λ1=10;λ2=10;λ3=10;λ4=10). The results for this example are shown in Table2(row Case1).These results conﬁrm the fact that most of the time is spent in module M3,which is the most computationally demanding.Moreover,module M1(respectively M4) spends most of its time waiting to send data on s1(respectively waiting to receive data from s4).M2is computing quickly,and this module is often receiving/sending from stream s2/s3(little time spent waiting on these streams in comparison with streams s1/s4).If we study the computational rate,we can thus decide to map M3alone on a powerful computing site because it has the highest value between the diﬀerent steady states probabilities of the modules.One should be careful to map the streams s1 and s4onto suﬃciently fast network links to increase the overall throughput of the network.A mapping that performs well can thus be deduced from this information, by adjusting the reasoning to the architecture of the available system. Experiment2.We can reproduce the same experiment but for a diﬀerent ap-plication:one in which there are a lot of data to be transfered inside the loop. Here,for one input on s1,the module M2makes several calls to the server M3 for computations.In this case,the rates of the streams are diﬀerent,for instance λ1=λ4=1000andλ2=λ3=1.The results for this experiment are shown in Table2(row Case2).In this table, we can see that M3is quite idle,waiting to receive data89.4%of the time(i.e.this is the time it is not processing).Moreover,we can see in the stream results that s2 and s3are busier than the other streams.In this case a good solution might be to map M2and M3on to the same cluster,since M3is no longer the computational bottleneck.We could thus have fast communication links for s2and s3,which are demanding a lot of network resources.Table2:Performance results for the example.Modules StreamsM1M2M3M4s1s2s3s4 Case1 4.2 5.167.0 4.247.0 6.7 6.747.0Case252.152.210.652.1 5.210.610.6 5.2Automatic mapping of ASSIST applications using process algebra 4.5.Analysis summaryAs mentioned in Section4.3,PEPA rates model both aspects strictly related to the mapping and to the application’s logical structure(such as algorithms imple-mented in the modules,communication patterns and size).The predictive analysis conducted in this work provides performance results which are related only to the application’s logical behavior.On the PEPA model this translates on the assump-tion that all sites includes platforms with the same computing power,and all links have an uniform speed.In other words,we assume to deal with a homogeneous grid to obtain the relative requirements of power among links and platforms.This information is used as a hint for the mapping on a heterogeneous grid.It is of value to have a general idea of a good mapping solution for the application, and this reasoning can be easily reﬁned with new models including the mapping peculiarities,as demonstrated in our previous work[1].However,the modeling technique exposed in the present paper allows us to highlight individual resources (links and processors)requirements,that are used to label the application graph.These labels represent the expected relative requirements of each module(stream) with respect to other modules(streams)during the application run.In the case of a module the described requirement can be interpreted as the aggregate power of the site on which it will be mapped.On the other hand,a stream requirement can be interpreted as the bandwidth of the network link on which it will be mapped.The relative requirements of parmods and streams may be used to implement mapping heuristics which assign more demanding parmods to more powerful sites,and more demanding streams to links exhibiting higher bandwidths.When a fully automatic application mapping is not required,modules and streams requirements can be used to drive a user-assisted mapping process.Moreover,each parmod exhibits a structured parallelism pattern(a.k.a.skele-ton).In many cases,it is thus possible to draw a reliable relationship between the site fabric level information(number and kind of processors,processors and network benchmarks)and the expected aggregate power of the site running a given parmod exhibiting a parallelism pattern[5,4,8].This may enable the development of a map-ping heuristic,which needs only information about sites fabric level information, and can automatically derive the performance of a given parmod on a given site.The use of models taking into account both of the system architecture charac-teristics can then eventually validate this heuristic,and give expected results about the performance of the application for a speciﬁed mapping.4.6.Future workThe approach described here considers the ASSIST modules as blocks and does not model the internal behavior of each module.A more sophisticated approach might be to consider using known models of individual modules and to integrate these with the global ASSIST model,thus providing a more accurate indication of the performance of the application.At this level of detail,distributed shared。

保积3-Hom-李超代数的积结构

第39卷第5期齐齐哈尔大学学报（自然科学版） Vol.39,No.5 2023年9月Journal of Qiqihar University(Natural Science Edition)Sep.,2023保积3-Hom-李超代数的积结构关宝玲1，田馨心2，田丽军3，闫煦1，王春艳1，李艳君1，董红月1，汪禹淼1，蒋加欣1，赵芬芬1（1.齐齐哈尔大学理学院，黑龙江齐齐哈尔 161006；2.中山大学数学学院，广州 510275；3.齐齐哈尔大学通信与电子工程学院，黑龙江齐齐哈尔 161006）摘要：把3-李代数的积结构推广到3-Hom-李超代数。

引入了保积3-Hom-李超代数上的积结构定义，给出了积结构存在的充分必要条件，得到一种特殊的积结构，严格积结构。

关键词：保积3-Hom-李超代数；积结构；严格积结构中图分类号：O152.5 文献标志码：A 文章编号：1007-984X(2023)05-0091-04Hom-李代数的概念是由HARTWIG 等[1]介绍的。

Hom-代数的结构最早出现在向量域上的李代数的拟形变和离散化领域，这些拟形变产生Hom-李代数，它是一个广义的李代数，它包含斜对称性和扭的Jacobi 等式。

Witt 代数和Virasoro 代数的一些q -形变中也有一个Hom-李代数的结构，它与离散的、形变的向量域和微积分学有着紧密联系[1-3]。

Hom-结构被引入到许多代数中，例如，Hom-李代数、Hom-结合代数、Hom-余代数、Hom-双代数、n -元Hom-Nambu-李代数、Hom-莱布尼兹代数和Hom-n -李超代数等等[4-11]。

三李代数最早哈密顿力学的NAMBU 推广[12]，三李超代数最早是由CANTARINI 介绍的[13]，3-Hom-李超代数是三李超代数的Hom-型推广。

GUAN 等[14]研究了3-Hom-李超代数的构造和诱导表示。

maple函数大全

rtable data structures
序包
Matlab
Matlab Link
与 Matlab 的接口函数
Ore_algebra
Bacic calculations in algebras of 线性算的基本代数运算
linear onerators
PDEtools
tools for solveing partial differential 偏微分方程相关函数
雅可比椭圆函数
雅可比椭圆函数雅
雅可比θ函数
JacobiZeta KelvinBei KelvinBer KelvinHei KelvinHer
雅可比ζ函数开尔文 Bei 函数开尔文 Her 函数开尔文 Hei 函数开尔文 Her 函数
KelvinKei KelvinHer KummerM KummerU LanbertW Lcm LegendreP LegendreQ Lerchphi Li Linsolve Lommels1 Lommels2 MOLS Maple_floats MatlabMatrix MeijerG 函数名称 Normal Nullspace Power Powmod Prem Primitive Primpart 函数名称
多项式可分解为同阶因式的可能性 Product（乘）函数的简化形式双γ函数与多γ函数 Quo（求多项式的商）函数的简化形式一种递归方程解的数据结构有限域的随机多项式有限域的随机 monic prime 多项式 Ratrecon（重组有理函数）的简化形式取复数的实部 rem（求多项式相除余项）函数的简化形式 Resultant（合成多项式）函数简化形式求方程的根多项式求模后的根 Sperm(异伪余式）函数的简化形式查找文本双曲正弦函数正弦积分矩阵的 Smith 正交化非平方因式分解转移正弦积分 StruveH 函数 StruveL 函数 sum(求和）和函数的简化形式基本功能计算矩阵的奇异数/向量显示文本

Kronecker product - Wikipedia, the free encyclopedia

Kronecker productFrom Wikipedia, the free encyclopediaIn mathematics, the Kronecker product, denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix. It is a generalization of the outer product (which is denoted by the same symbol) from vectors to matrices, and gives the matrix of the tensor product with respect to a standard choice of basis. The Kronecker product should not be confused with the usual matrix multiplication, which is an entirely different operation.The Kronecker product is named after Leopold Kronecker, even though there is little evidence that he was the first to define and use it. Indeed, in the past the Kronecker product was sometimes called the Zehfuss matrix, after Johann Georg Zehfuss.Contents1 Definition1.1 Examples2 Properties2.1 Relations to other matrix operations2.2 Abstract properties3 Matrix equations4 Related matrix operations4.1 Tracy-Singh product4.2 Khatri-Rao product5 See also6 Notes7 References8 External linksDefinitionIf A is an m × n matrix and B is a p × q matrix, then the Kronecker product A⊗B is the mp × nq block matrix:more explicitly:If A and B represent linear transformations V1 → W1 and V2 → W2, respectively, then A⊗B represents the tensor product of the two maps, V1⊗V2 → W1⊗W2.ExamplesPropertiesRelations to other matrix operations1. Bilinearity and associativity: The Kronecker product is a special case of the tensorproduct, so it is bilinear and associative:where A, B and C are matrices and k is a scalar.2. Non-commutative: In general A⊗B and B⊗A are different matrices. However, A⊗Band B⊗A are permutation equivalent, meaning that there exist permutation matrices P and Q such thatIf A and B are square matrices, then A⊗B and B⊗A are even permutation similar,meaning that we can take P = Q T.3. The mixed-product property and the inverse of a Kronecker product: If A, B, C andD are matrices of such size that one can form the matrix products AC and BD, thenThis is called the mixed-product property, because it mixes the ordinary matrix product and the Kronecker product. It follows that A⊗B is invertible if and only if A and B are invertible, in which case the inverse is given by4. Transpose: The operation of transposition is distributive over the Kronecker product:5. Determinant: Let A be an n × n matrix and let B be a p × p matrix. ThenThe exponent in |A| is the order of B and the exponent in |B| is the order of A.6. Kronecker sum and exponentiation If A is n × n, B is m × m and I k denotes the k ×k identity matrix then we can define what is sometimes called the Kronecker sum, ⊕, byNote that this is different from the direct sum of two matrices. This operation isrelated to the tensor product on Lie algebras. We have the following formula for thematrix exponential which is useful in the numerical evaluation of certain continuous-time Markov processes[citation needed],Kronecker sums appear naturally in physics when considering ensembles of non-interacting systems. Let H i be the Hamiltonian of the i-th such system. Then the total Hamiltonian of the ensemble is.Abstract properties1. Spectrum: Suppose that A and B are square matrices of size n and m respectively. Letλ1, ..., λn be the eigenvalues of A and μ1, ..., μm be those of B (listed according to multiplicity). Then the eigenvalues of A⊗B areIt follows that the trace and determinant of a Kronecker product are given by2. Singular values: If A and B are rectangular matrices, then one can consider theirsingular values. Suppose that A has r A nonzero singular values, namelySimilarly, denote the nonzero singular values of B byThen the Kronecker product A⊗B has r A r B nonzero singular values, namelySince the rank of a matrix equals the number of nonzero singular values, we find that3. Relation to the abstract tensor product: The Kronecker product of matricescorresponds to the abstract tensor product of linear maps. Specifically, if the vector spaces V, W, X, and Y have bases {v1, ..., v m}, {w1, ..., w n}, {x1, ..., x d}, and {y1,..., y e}, respectively, and if the matrices A and B represent the linear transformations S : V → X and T : W → Y, respectively in the appropriate bases, then the matrix A⊗B represents the tensor product of the two maps, S⊗T : V⊗W → X⊗Y with respect to the basis {v1⊗ w1, v1⊗ w2, ..., v2⊗ w1, ..., v m⊗ w n} of V⊗W and the similarlydefined basis of X⊗Y with the property that A⊗B(v i⊗ w j) = (A v i)⊗(B w j), where i and j are integers in the proper range.[1] When V and W are Lie algebras, and S : V → V and T : W → W are Lie algebra homomorphisms, the Kronecker sum of A and B represents theinduced Lie algebra homomorphisms V⊗W → V⊗W.4. Relation to products of graphs: The Kronecker product of the adjacency matrices oftwo graphs is the adjacency matrix of the tensor product graph. The Kronecker sum of the adjacency matrices of two graphs is the adjacency matrix of the Cartesian product graph.See,[2] answer to Exercise 96.Matrix equationsThe Kronecker product can be used to get a convenient representation for some matrix equations. Consider for instance the equation AXB = C, where A, B and C are given matrices and the matrix X is the unknown. We can rewrite this equation asHere, vec(X) denotes the vectorization of the matrix X formed by stacking the columns of Xinto a single column vector. It now follows from the properties of the Kronecker product that the equation AXB = C has a unique solution if and only if A and B are nonsingular (Horn & Johnson 1991, Lemma 4.3.1).If X is row-ordered into the column vector x then AXB can be also be written as (Jain 1989, 2.8 Block Matrices and Kronecker Products) (A⊗B T)x.Related matrix operationsTwo related matrix operations are the Tracy-Singh and Khatri-Rao products which operate on partitioned matrices. Let the m × n matrix A be partitioned into the m i × n j blocks A ij and p × q matrix B into the p k × qℓ blocks B kl with of course Σi m i = m, Σj n j = n, Σk p k = p and Σℓ qℓ = q.Tracy-Singh productThe Tracy-Singh product[3][4] is defined aswhich means that the (ij)-th subblock of the mp × nq product A ○ B is the m i p × n j q matrix A ij ○ B, of which the (kℓ)-th subblock equals the m i p k × n j qℓ matrix A ij⊗B kℓ. Essentially the Tracy-Singh product is the pairwise Kronecker product for each pair of partitions in the two matrices.For example, if A and B both are 2 × 2 partitioned matrices e.g.:we get:Khatri-Rao productThe Khatri-Rao product[5][6] is defined asin which the ij-th block is the m i p i × n j q j sized Kronecker product of the corresponding blocks of A and B, assuming the number of row and column partitions of both matrices is equal. The size of the product is then (Σi m i p i) × (Σj n j q j). Proceeding with the same matrices as the previous example we obtain:This is a submatrix of the Tracy-Singh product of the two matrices (each partition in this example is a partition in a corner of the Tracy-Singh product).A column-wise Kronecker product of two matrices may also be called the Khatri-Rao product.This product assumes the partitions of the matrices are their columns. In this case m1 = m, p1 = p, n = q and for each j: n j = p j = 1. The resulting product is a mp × n matrix of which each column is the Kronecker product of the corresponding columns of A and B. Using the matrices from the previous examples with the columns partitioned:so that:See alsoGeneralized linear array modelMatrix productNotes1. ^ Pages 401–402 of Dummit, David S.; Foote, Richard M. (1999), Abstract Algebra (2 ed.), New York:John Wiley and Sons, Inc., ISBN 0-471-36857-12. ^ D. E. Knuth: "Pre-Fascicle 0a: Introduction to Combinatorial Algorithms" (http://www-cs-/~knuth/fasc0a.ps.gz), zeroth printing (revision 2), to appear as part of D.E.Knuth: The Art of Computer Programming Vol. 4A3. ^ Tracy, DS, Singh RP. 1972. A new matrix product and its applications in matrix differentiation.Statistica Neerlandica 26: 143–157.4. ^ Liu S. 1999. Matrix results on the Khatri-Rao and Tracy-Singh products. Linear Algebra and itsApplications 289: 267–277. (pdf (/science?_ob=MImg&_imagekey=B6V0R-3YVMNR9-R-1&_cdi=5653&_user=877992&_orig=na&_coverDate=03%2F01%2F1999&_sk=997109998&view=c&wchp=dGLbVlb-zSkWb&md5=21c8c66f17da8d1bab45304a29cc96ac&ie=/sdarticle.pdf))5. ^ Khatri C. G., C. R. Rao (1968), "Solutions to some functional equations and their applications tocharacterization of probability distributions" (http://sankhya.isical.ac.in/search/30a2/30a2019.html), Sankhya30: 167–180.6. ^ Zhang X, Yang Z, Cao C. (2002), "Inequalities involving Khatri-Rao products of positive semi-definite matrices", Applied Mathematics E-notes2: 117–124.ReferencesHorn, Roger A.; Johnson, Charles R. (1991), Topics in Matrix Analysis, CambridgeUniversity Press, ISBN 0-521-46713-6.Jain, Anil K. (1989), Fundamentals of Digital Image Processing, Prentice Hall, ISBN 0-13-336165-9.Steeb, Willi-Hans (1997), Matrix Calculus and Kronecker Product with Applications and C++ Programs, World Scientific Publishing, ISBN 981-02-3241-1Steeb, Willi-Hans (2006), Problems and Solutions in Introductory and Advanced MatrixCalculus, World Scientific Publishing, ISBN 981-256-916-2External linksHazewinkel, Michiel, ed. (2001), "Tensor product"(/index.php?title=p/t092410), Encyclopedia ofMathematics, Springer, ISBN 978-1-55608-010-4Kronecker product (/?op=getobj&from=objects&id=4163),.MathWorld Kronecker Product (/KroneckerProduct.html)New Kronecker product problems (http://issc.uj.ac.za/downloads/problems/newkronecker.pdf) Earliest Uses: The entry on The Kronecker, Zehfuss or Direct Product of matrices hashistorical information. (/k.html)Generic C++ and Fortran 90 codes for calculating Kronecker products of two matrices.(https:///projects/kronecker/)Retrieved from "/w/index.php?title=Kronecker_product&oldid=556239113" Categories: Matrix theoryThis page was last modified on 22 May 2013 at 09:24.Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy.Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization.。

Kronecker product - Wikipedia, the free encyclopedia

Bilingual Education

Anti-Bilingual Initiative "Babel" in SchoolsBilingual BallotsBilingual EducationBilingual ResearchCanardsDemographic Change Endangered LanguagesEnglish OnlyEnglish PlusLanguage RightsLanguage LegislationLife After Prop. 227LouisianaMuhlenberg Legend "Multilingual Government" National IdentityOfficial EnglishOpinion PollsPuerto RicoResearch ControversyRecent MonographsJ im Cummins, Beyond Adversarial Discourse (1998)Jay Greene, A Meta-Analysis of the Effectiveness of Bilingual Education (1998)Stephen Krashen, A Note on Greene's "Meta-Analysis of the Effectiveness of Bilingual Education"(1998)Kenji Hakuta, Memorandum on Read Institute press release Issues in U.S. Language Policy Bilingual EducationFor many Americans, bilingual education seems to defy common sense – not to mention the Melting Pot tradition. They ask:∙If non-English-speaking students are isolated in foreign-language classrooms,how are they ever going to learn English,the key to upward mobility?∙What was wrong with the old "sink or swim" method that worked forgenerations of earlier immigrants?∙Isn't bilingual education just another example of "political correctness" runamok –the inability to say no to avociferous ethnic lobby?Some English Only advocates go further, arguing that even if bilingual education is effective –which they doubt –it's still a bad idea for the country because bilingualism threatens to sap our sense of national identity and divide us along ethnic lines. They fear that any government recognition of minority languages "sends the wrong message" to immigrants, encouraging them to believe they can live in the U.S.A. without learning English or conforming to "American" ways.on NRC Report (1997)Diane August and Kenji Hakuta, Letter to Rosalie Porter(1997)Such complaints have made bilingual education a target of political attacks. One of the most serious to date is now under way in California,a ballot initiative that wouldmandate English-only instruction for all children until they become fully proficient in English.No doubt many of the objections to bilingual education are lodged in good faith. Others reflect ethnic stereotypes or class biases. Sad to say, they all reflect a pervasive ignorance about how bilingualeducation works, how second languages are acquired, and how the nation has responded to non-English-speaking groups in the past.Reinforcing popular fallacies requires less space thandeconstructing them. That's why my writing on these issues grew from a handful of newspaper articles into a 310-page book, Bilingual Education: History, Politics, Theory, and Practice (3rd ed., 1995).Nevertheless, a few points:Science is often counterintuitive. Its breakthroughs tend to upsetcommon-sense notions, not to mention cherished myths. Linguistics is no exception. In fact, it invites more than its share of opposition from nonspecialists – witness the"Ebonics" controversy – because most people feel like experts when it comes to language. Our reactions are often visceral. Perhaps that's because our speech defines us ethnically,socially, and intellectually. It's tied up with a sense of who we are –and who we are not – evoking some of our deepest emotions.What once seemed obvious aboutbilingualism – for example, that it handicaps children's cognitive growth – has usually proved unfounded. Since the 1960s, research has shown that multiple language skills do not confuse the mind. Quite the contrary: when well-developed, they seem to provide cognitive advantages, although such effects are complex and difficult to measure (Hakuta 1986).Another discredited notion is thatchildren will "pick up" a second language rapidly if "totally immersed" in it. For generations, this philosophy served to justify policies of educational neglect –assigning minority students to regular classrooms, with no special help in overcoming language barriers. Disproportionate numbers failed and dropped out of school as a result. The sink-or-swim approach was ruled illegal by the U.S. Supreme Court in Lau v. Nichols (1974).Research has shown that the quality– not the quantity – of English exposure is the major factor in English acquisition. That is, the second-language input must be comprehensible(Krashen 1996). Otherwise, it's just noise.English as a second language (ESL)is best taught in natural situations, with the second language used in meaningful contexts rather than in repetitious drills of grammar and vocabulary. One variant of ESL, known as "sheltered subject-matter instruction," adapts lessons to students' level of English proficiency. This approach is common in bilingual education programs, coordinated with lessons in students' native language.Native-language instruction alsohelps to make English comprehensible, by providing contextual knowledge that aids in understanding. When children already know something about dinosaurs, a lesson on the subject will make more sense when instruction shifts to English. Not only will they learn more about dinosaurs; they will also acquire more English.The same principle applies when itcomes to acquiring literacy. Teaching in the native language can facilitate the process, as the linguist Stephen Krashen (1996) explains:∙We learn to read by reading, by making sense of what we see on the page. ...∙If we learn to read by reading, it will be much easier to learn to read in a languagewe already understand.∙Once you can read, you can read. The ability to read transfers across languages."Language is not a unitary skill, but a complex configuration of abilities" (Hakuta and Snow1986). Social communication skills – a.k.a. playground English–should not be confused with academic English,the cognitively demanding language that children need to succeed in school. While playground English tends to be acquired rapidly by most children, academic English is typically acquired over a period of five to seven years (Cummins 1989).Research on the effectiveness ofbilingual education remains in dispute, because program evaluation studies – featuring appropriate comparison groups and random assignment of subjects or controls for pre-existing differences – are extremely difficult to design. Moreover, there is considerable variation among the pedagogies, schools, students, and communities being compared. While numerous studies have documented the benefits of bilingual programs, much of this research has faced methodological criticisms – as noted by an expert panel of the National Research Council (August and Hakuta 1997a).Certain critics of bilingualeducation have interpreted the NRC report to mean that, despite a generation of research, "there is no evidence that there will be long-term advantages or disadvantages to teaching limited-English students in the native language" (Glenn 1997). This conclusion – widely circulated by the so-called READ Institute –has been rejected by the NRC study directors.To the contrary, they say, the expert panel concluded that "agreat deal has been learned from the research that has been conducted on English language learners." Moreover, there are "empirical results . . . support[ing] the theory underlying native language instruction" (August and Hakuta 1997b). According to the panel's chairman, the "attempt by READ to place its own political spin" on the report hardly advances the cause of objective research (Hakuta 1997).Other critics continue to deny thatsuch empirical support exists. A recent "review of the literature" (Rossell and Baker1996) reports that bilingual education is inferior to English-only programs of all kinds, including sink-or-swim. Yet these conclusions owe more to the manipulation of program labels than to student performance in the classroom. Critiques of Rossell and Baker by Cummins (1998) and Krashen (1996) show that, among other distortions, the researchers rely heavily on studies of French immersion in Canada – bilingual or trilingual approaches that they portray as monolingual "immersion" or "submersion" models. Meanwhile, a meta-analysis of the same body of research reviewed by the critics, but using a more rigorous methodology, found quite different results: a significant edge for bilingual education (Greene 1998).The most sophisticated evaluationstudy to date – a four-year, longitudinal study of 2,000Spanish-speaking students in five states –found that "late-exit," developmental bilingual programs proved superior to "early-exit," transitional bilingual andEnglish-only immersion programs (Ramírez et al. 1991). That is, in programs that stressednative-language skills, students' growth in English reading and mathematics continued to increase long after it had leveled off among their peers in the other programs. While this study has been praised by many, others have rejected the comparison as invalid because all three programs were not tested in the same school districts.Nevertheless, a consensus ofapplied linguists recognizes that the following propositions have strong empirical support:∙Native-language instruction does not retard the acquisition of English.∙Well-developed skills in the native language are associated with high levelsof academic achievement.∙Bilingualism is a valuable skill, for individuals and for the country.Bilingual education was adopted by manylocal school districts in the 1960s and 1970s to remedy practices that had denied language minorities an equal educational opportunity. Yet it was hardly a new invention designed to replace the Melting Pot with the Salad Bowl or some other model of ethnic pluralism. There is a long bilingual tradition in the U.S.A., in which minority-language schooling has played a central, albeit largely forgotten, role.seeRivera Center.Hakuta, Kenji. 1997. Memorandum on Read Institute press release on NRC Report.Hakuta, Kenji. 1986. Mirror of Language: The Debate on Bilingualism.New York: Basic Books.Hakuta, Kenji, and Snow, Catherine. 1986. "The Role of Research in Policy Decisions about Bilingual Education." NABE News 9, no. 3 (Spring): 1, 18-21.Krashen, Stephen D. 1996. Under Attack: The Case Against Bilingual Education. Culver City, Calif.: Language Education Associates.Ramírez, J. David; Yuen, Sandra D.; and Ramey, Dena R. 1991. Final Report: Longitudinal Study of Structured Immersion Strategy, Early-Exit, and Late-Exit Transitional Bilingual Education Programs forLanguage-Minority Children.San Mateo, Calif.: Aguirre International.Rossell, Christine, and Baker, Keith. 1996. The effectiveness of bilingual education. Research in the Teaching of English,30,pp.7-74.Copyrigh t © 1998 by James Crawford. Permission is hereby granted to reproduce this page for free, noncommercial distribution, provided that credit is given and this notice is included. Requests for。

带有不变双线性型的Zinbiel代数

（２）
则称Ａ是一个Ｚｉｎｂｉｅｌ代数．
对于Ｚｉｎｂｉｅｌ代数Ａ来说，反交换子ｘｙ＝ｘｙ＋

ｙｘ定义了一个交换结合代数，称为Ｚｉｎｂｉｅｌ代数Ａ的
结合代数．
令Ｌ（ｘ）、Ｒ（ｘ）分别表示Ｚｉｎｂｉｅｌ代数Ａ的左乘
Ｌｅｉｂｎｉｚ代数［１］作为李代数的推广形式受到广泛关注，Ｚｉｎｂｉｅｌ代数是与Ｌｅｉｂｎｉｚ代数有着所谓Ｋｏｓｚｕｌ对偶关系的代数［２］，又被称为对偶Ｌｅｉｂｎｉｚ代数．Ｚｉｎｂｉｅｌ代数关于反交换子ａｂ＝ａｂ＋ｂａ是一个交换结合代数，在代数范畴定义了一种算子（Ｚｉｎｂｉｅｌ）→ （Ｃｏｍ），是函子（Ｌｉｅ）→（Ｚｉｎｂｉｅｌ）的对偶［３－４］．
第３期
于建华：带有不变双线性型的Ｚｉｎｂｉｅｌ代数
９５
ｆ（ｘｙ，ｚ）＋ｆ（ｙ，ｘｚ）＝０（ｘ，ｙ，ｚＡ）．（１）
下面给出Ｚｉｎｂｉｅｌ代数上不变双线性型的例子．
例１设Ａ是４维Ｚｉｎｂｉｅｌ代数．关于基ｅ１，ｅ２，ｅ３，ｅ４，Ａ的乘积如下：ｅ１ｅ２＝ｅ３，ｅ１ｅ３＝ｅ４，ｅ２ｅ１＝－ｅ３，ｅ２ｅ２＝ｅ４，其余为零，则Ａ上的不变双线性型ｆ的度量矩阵为
１预备知识
不变双线性型在几何上密切联系伪黎曼度量，类似于李代数，给出双线性型“不变”的定义：
定义１一个双线性型ｆ：Ａ×Ａ→称为是不变的，如果满足
收稿日期：２０１７－０４－２６《华南师范大学学报（自然科学版）》网址：ｈｔｔｐ：／／ｊｏｕｒｎａｌ．ｓｃｎｕ．ｅｄｕ．ｃｎ／ｎ基金项目：国家自然科学基金项目（１１６７１１５３）；广东省自然科学基金项目（２０１６Ａ０３０３１３８５０）通讯作者：于建华，讲师，Ｅｍａｉｌ：ｙｕｊｉａｎｈｕａｓｃｎｕ＠１２６．ｃｏｍ．

HOPF ALGEBRAS

uA
θ ⊗θ
/ B⊗B
θ
/B O
uB
k commute.
k
Deﬁnition. Let C and D be k -coalgebras. A linear map f : C → D is a k -coalgebra morphism if the diagrams C C ⊗C and C
HOPF ALGEBRAS
5
For example, if A is any associative k -algebra then A is a Lie algebra if we deﬁne [a, b] = ab − ba for all a, b ∈ A. In particular, when A = Mn (k ), we write this as gl(n, k ). The universal enveloping algebra of the Lie algebra g is the factor algebra of the tensor algebra T (g) by the ideal I (g) =< [x, y ] − xy + yx : x, y ∈ g > There is a bijective correspondence between left U (g)-modules and representations of the Lie algebra g, where the latter are - by deﬁnition - Lie algebra homomorphisms ρ : g → End(V ) for k -vector spaces V . Special case (A): Let n ≥ 1, g =

BasicAlgebra2

Basic Algebra 2We know when students are just beginning their study of algebra, the basic concepts can be difficult to grasp. These activities are written for teachers to use in their basic algebra classes.When you find other helpful exercises, add these to your own eActivities.Good exercises encourage students!This file includes eActivities on:1 Variable Calculation – Substitute the number to the variable.2 Expression with Variable – Expand the expression, checking your work with Verify.3 Product 1 – Find the product of the examples.4 Product 2 – Verify will help keep you on the right track.5 Proportion – Do you know how to solve for x when dealing with proportions?6 Factoring – Practice factoring with Verify.7 Factoring & Root – Utilize Verify along with Analysis/G-Solve/Root in the Graph application.8 Polynomial Remainder – Find the remainder in these examples.1 Variable CalculationSubstitute the number to the variable..2 Expression with VariableExpand the expression, checking your work with Verify.3 Product 1Find a product using the distributive property.4 Product 2Verify will help keep you on the right track.5 ProportionDo you know how to solve for x when dealing with proportions?6 FactoringPractice factoring with Verify.7 Factoring and RootUtilize Verify along with Analysis/G-Solve/Root in the Graph application.8 Polynomial (Remainder) Find the remainder in these examples.。

ElementaryLinearAlgebraInstructorSolutionsManual

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On Lie 2-bialgebras

索罗干-班东甘辛学习模型教程说明书

Algebra_II_Honors_Syllabus_2012-2013

Generalized WDVV equations for B_r and C_r pure N=2 Super-Yang-Mills theory

differential algebras

一类超Heisenberg-Virasoro代数的超双代数 结构

Combinatorial Hopf algebras and Towers of Algebras

Exponential map - Wikipedia, the free encyclopedia

pacs码

Received (received date) Revised (revised date)

保积3-Hom-李超代数的积结构

maple函数大全

Kronecker product - Wikipedia, the free encyclopedia

Kronecker product - Wikipedia, the free encyclopedia

Bilingual Education

带有不变双线性型的Zinbiel代数

HOPF ALGEBRAS

BasicAlgebra2

ElementaryLinearAlgebraInstructorSolutionsManual

一类超Heisenberg-Virasoro代数的超双代数结构