On simple modules for certain pointed Hopf algebras [electronic resource] /
Abstract (Summary)In 2003, Radford introduced a new method to construct simple modules for the Drinfel'd double of a graded Hopf algebra. Until then, simple modules for such algebras were usually constructed by taking quotients of Verma modules by maximal submodules. This new method gives a more explicit construction, in the sense that the simple modules are given as subspaces of the Hopf algebra and one can easily find spanning sets for them. I use this method to study the representations of two types of pointed Hopf algebras: restricted two-parameter quantum groups, and theDrinfel'd double of rank one pointed Hopf algebras of nilpotent type. The groups of group-like elements of these Hopf algebras are abelian; hence, they fall among those Hopf algebras classified by Andruskiewitsch and Schneider. I 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 a special quotient. For restricted two-parameter quantum groups, given [theta] a primitive ?th root of unity, the factorization of simple [unable to replicate formula]-modules is possible, if and only if gcd((y-z)n, ?) = 1. I construct simple modules using the computer algebra system Singular::Plural and present computational results and conjectures about bases and dimensions. For rank one pointed Hopf algebras, given the data D = [unable to replicate formula], the factorization of simple D(H[subscript] D)-modules is possible if and only if [unable to replicate formula] is odd and [unable to replicate formula]. Under this condition, the tensor product of two simple D(H[subscript]D)-modules is completely reducible, if and only if the sum of their dimensions is less or equal than [unable to replicate formula].
School Location:USA - Texas
Source Type:Master's Thesis
Keywords:major mathematics hopf quantum
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