Factorization of simple modules for certain pointed Hopf algebras

[摘要] We study the representations of two types of pointed Hopf algebras: restricted two-parameter quantum groups, and the Drinfel'd doubles of rank one pointed Hopf algebras of nilpotent type. We study, in particular, under what conditions a simple module can be factored as the tensor product of a one-dimensional module with a module that is naturally a module for the quotient by central group-like elements. For restricted two-parameter quantum groups, given theta a primitive lth root of unity, the factorization of simple u(theta y,theta z) (sl(n))-modules is possible, if and only if gcd((y-z)n, l) = 1. For rank one pointed Hopf algebras, given the data D (G, X, a), the factorization of simple D(H-D)-modules is possible if and only if |chi (a)| is odd and |chi (a)| |a| = |X|. Under this condition, the tensor product of two simple D(H-D)-modules is completely reducible, if and only if the sum of their dimensions is less than or equal to |X (a)| + 1. (C) 2007 Elsevier Inc. All rights reserved.

[发布日期] 2007-12-15

[关键词] Hopf algebras;Drinfel'd double;quantum groups