On simple modules for certain pointed Hopf algebras

Description

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 the
Drinfel’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 θ a primitive ℓth root
of unity, the factorization of simple uθy,θz (sln)-modules is possible, if and only if
gcd((y − z)n, ℓ) = 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, χ, a),
the factorization of simple D(HD)-modules is possible if and only if |χ(a)| is odd and
|χ(a)| = |a| = |χ|. 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 |χ(a)| + 1.

Links & Downloads

1. http://hdl.handle.net/1969.1/4683

Tags

Hopf

quantum

Additional Fields

Identifer	oai:union.ndltd.org:tamu.edu/oai:repository.tamu.edu:1969.1/4683
Date	25 April 2007
Creators	Pereira Lopez, Mariana
Contributors	Witherspoon, Sarah
Publisher	Texas A&M University
Source Sets	Texas A and M University
Language	en_US
Detected Language	English
Type	Book, Thesis, Electronic Dissertation, text
Format	329988 bytes, electronic, application/pdf, born digital