Title

Representations of semisimple Hopf algebras

Date of Award

2005

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

Advisor(s)

Declan P. Quinn

Keywords

Hopf algebras, Drinfeld double, Semisimple

Subject Categories

Mathematics | Physical Sciences and Mathematics

Abstract

Let H be a cosemisimple Hopf algebra over an algebraically closed field. In the first chapter of the thesis, it is shown that if H has a simple subcoalgebra of dimension 9 and has no simple subcoalgebras of even dimension, then H contains either a grouplike element of order 2 or 3, or a family of simple subcoalgebras whose dimensions are the squares of each positive odd integer. In particular, if H is odd dimensional, then its dimension is divisible by 3.

In the second chapter, the induced representations from H and H * to the Drinfel'd double D ( H ) are studied. The product of two such representations is a sum of copies of the regular representation of D ( H ). The action of certain irreducible central characters of H * on the simple modules of H is considered. The modules that receive trivial action from each such irreducible central character are precisely the constituents of the tensor powers of the adjoint representation of H .

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