Representations of semisimple Hopf algebras
Date of Award
Doctor of Philosophy (PhD)
Declan P. Quinn
Hopf algebras, Drinfeld double, Semisimple
Mathematics | Physical Sciences and Mathematics
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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Burciu, Sebastian, "Representations of semisimple Hopf algebras" (2005). Mathematics - Dissertations. 29.