General+Double+Quantum+Groups

GENERAL DOUBLE QUANTUM GROUPS
Tianshui Ma and Shuanhong Wang
Department of Mathematics, Southeast University, Nanjing, Jiangsu, China
We introduce the concept of a twisted tensor biproduct, generalizing the general product given in Caenepeel et al. [1]. We give necessary and sufficient conditions for the new object to be a bialgebra and study the case of ∗-bialgebras. We describe all braided structures on the twisted tensor biproduct and introduce the notion of a general double quantum group. As an application, the special cases and examples are given. Key Words: Braided Hopf algebra; Braided monoidal category; General double quantum group; Twisted tensor biproduct. 2000 Mathematics Subject Classification: 16W30.
646
MA AND WANG
In Section 2, we study the twisted tensor biproduct AT R B, where B is a bialgebra and A is only an algebra and a coalgebra connected by a twisted map R B ⊗ A −→ A ⊗ B and a cotwisted map T A ⊗ B −→ B ⊗ A. We find in Theorem 2.1 necessary and sufficient conditions for the new object AT R B to be a bialgebra and study in Theorem 2.3 the case of ∗-bialgebras. In Section 3, we describe in Theorem 3.12 all braided structures on the twisted tensor biproduct AT R B. We study the relations between the existence of braided structures on A and B and the ones on AT R B (see Theorem 3.13). Finally, in Section 4 we conclude by describing its applications and examples. 1. PRELIMINARIES Throughout, k is the fixed field, and is the complex numbers field. Unless otherwise stated, all vector spaces are over k and all maps are k-linear. For vector spaces U and V over k, we drop the subscript k from U ⊗k V and Homk U V and write U ⊗ V and Hom U V instead. The linear map U V U ⊗ V −→ V ⊗ U defined by U V u ⊗ v = v ⊗ u for all u ∈ U and v ∈ V is the flip map. The identity map of U is denoted by iU . Our basic reference on Hopf algebras is Sweedler [15], on braided Hopf algebras Larson and Towber [7] and on braided monoidal category theory Maclane [8]. Let C be a coalgebra with comultiplication C −→ C ⊗ C . We follow the widely used convention of representing c ∈ C ⊗ C for c ∈ C symbolically by c = c1 ⊗ c2 , a notation that originates in the Heyneman–Sweedler notation c = c 1 ⊗ c 2 for the coproduct. In particular, the structure map of a left C -comodule V is denoted by v = v −1 ⊗ v0 for v ∈ V , and the structure map of a right C -comodule U is denoted by u = u0 ⊗ u 1 for u ∈ U . We write C for the category of left comodules over the coalgebra C , and A for the category of left modules over the algebra A. Recall that a ∗-algebra A over has an involution a → a∗ for all a ∈ A. An involution is an antilinear map satisfying a∗∗ = a and ab ∗ = b∗ a∗ for all a b ∈ A. If A B are ∗-algebras then the tensor product algebra A ⊗ B is again a ∗-algebra for the product defined as a ⊗ b a ⊗ b = aa ⊗ bb and the involution given by a ⊗ b ∗ = a∗ ⊗ b∗ for all a a ∈ A and b b ∈ B. Recall that a Hopf ∗-algebra A is a Hopf algebra A over with a coproduct A −→ A ⊗ A, with a counit A −→ and with an antipode S A −→ A so that and are ∗-homomorphisms and S S a ∗ ∗ = a for all a ∈ A, respectively. 1.1. The Twisted Tensor Product of ∗-Algebras Let A and B be algebras with units 1A and 1B , respectively. Suppose that R B ⊗ A −→ A ⊗ B is a linear map. The A#R B is defined to be a vector space A ⊗ B with the product given by mA#R B = mA ⊗ mB iA ⊗ R ⊗ iB or a#R b a #R b = aaR #bR b (1.1)
Received June 19, 2008; Revised April 13, 2009. Communicated by E. Puczylowski. Address correspondence to Prof. Shuanhong Wang, Department of Mathematics, Southeast University, Nanjing, Jiangsu 210096, China; E-mail: shuanhwang2002@ 645
INTRODUCTION Double quantum group is a braided (i.e., coquasitriangular) Hopf algebra. This construction originated independently in Podles and Woronowicz’s construction of the quantum Lorentz group (see Podles and Woronowicz [13]) and in early work of Majid on double cross products (see Majid [9]). As a generalization of Majid’s double cross products, a general product AT ⊗R B between two bialgebras A and B connected via a twisted map R B ⊗ A −→ A ⊗ B, and a cotwisted map T A ⊗ B −→ B ⊗ A was introduced in Caenepeel et al. [1]. The product construction AT ⊗R B is equipped with both smash product construction A#R B and smash coproduct construction A ×T B on A ⊗ B. In particular, the authors derived in Caenepeel et al. [1, Theorem 4.5] necessary and sufficient conditions for AT ⊗R B to be a bialgebra. In the above construction AT ⊗R B, we will consider in this article the case that B is a bialgebra and A is only an algebra and a coalgebra. The new product is called a twisted tensor biproduct denoted by AT R B, generalizing Radford’s bismash products in Radford [14]. The general double quantum group constructions on AT R B will be established. The article is organized as follows. In Section 1, we recall the definitions and some of the basic properties of twisted tensor (co)products, double quantum groups, and Yetter–Drinfeld categories, respectively.
Communications in Algebra® , 38: 645–672, 2010 Copyright © Taylor & Francis Group, LLC ISSN: 0092-7872 priห้องสมุดไป่ตู้t/1532-4125 online DOI: 10.1080/00927870903100077