On simple modules for certain pointed Hopf algebras
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Date
2007-04-25
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Texas A&M University
Abstract
In 2003, Radford introduced a new method to construct simple modules for
the DrinfelâÂÂd 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 the
DrinfelâÂÂd 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 ø a primitive âÂÂth root
of unity, the factorization of simple uøy,øz (sln)-modules is possible, if and only if
gcd((y â z)n, âÂÂ) = 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 = (G, ÃÂ, a),
the factorization of simple D(HD)-modules is possible if and only if |ÃÂ(a)| is odd and
|ÃÂ(a)| = |a| = |ÃÂ|. Under this condition, the tensor product of two simple D(HD)-modules is completely reducible, if and only if the sum of their dimensions is less or
equal than |ÃÂ(a)| + 1.
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Keywords
Hopf, quantum