Cosemisimple Hopf Algebras and the FRT Construction

Abstract

Since the quantum groups have been invented by Drinfel'd [5], they have mainly been considered from two directions: deformations (or quantizations) of universal enveloping algebras of complex semisimple Lie algebras and deformations of coordinate algebras of simple, simply-connected Lie groups. The latter algebras can either be investigated using the ingenious FRT construction [6], starting from solutions of the quantum Yang-Baxter equation, which can also deduced from the deformations of the universal enveloping algebras, or in a more abstract way as a certain Hopf algebra in the dual of the deformed universal enveloping algebra. The ®rst approach has been generalized in various ways, including [15,1,20,2]. An interesting problem is to ®nd out how these two approaches are related. The obtained algebras are equal if and only if a certain canonical map from the algebra into the dual of the quantized universal enveloping algebra is injective. In the non-quantized case, there is a general result in [17]. In the quantized case, an analogous duality has been proved by Takeuchi [19] for the Lie algebra sl n (cf. also [18]). For the orthogonal and symplectic Lie algebras, there are similar results (starting from [6] up to [11, Corollary 11.23]). These results are based on lengthy calculations. In fact the main reason why the two approaches are equivalent, is the so-called Brauer-Schur-Weyl duality. If some analogue of it is satis®ed, a similar description even holds in greater generality for cosemisimple coquasitriangular Hopf 4377 algebras which are generated by the matrix coecients of a ®nite-dimensional simple comodule. We prove some properties of certain distinguished elements of the deformations of coordinate algebras, e.g. the``quantum determinant'', and indicate some properties of some analogues of exterior and symmetric algebras in a rather general setting.After introducing notations, we prove the structural result for cosemisimple coquasitriangular Hopf algebras. Finally we apply this method to the above mentioned quantum groups. NOTATIONIn the following let Z, N, N 0 , C denote the sets of integers, positive integers, non-negative integers, and complex numbers, respectively. Fix a ®eld k. For each n P N let GL n k be the group of invertible n  n matrices over k. All objects we consider are k-vector spaces. All tensor products are over k. Any module will be a left module. Let V be a vector space. We denote by V à : Hom k V; k the dual space of V. For m P N 0 , we use V m V Á Á Á V to denote the tensor product of m copies of V (and V 0 : k). Similarly, if T : V 3 V is a linear operator, denote T m T Á Á Á T : V m 3 V m the tensor product of m copies of this operator. The tensor algebra TV of V is the coproduct of all vector spaces V m for m P N 0 with the tensor product as multiplication. It has a natural gradation given by TV m : V m for m P N 0 . If V has the linear basis v 1 ; . . . ; v n then TV is natural isomorphic to the free algebra generated by v 1 ; . . . ; v n . We use some index convention regardin...

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