On the algebra structure of some bismash products

Details On the algebra structure of some bismash products On the algebra structure of some bismash products

2011-08-06

arxiv.org/abs/1108.1512v1

Journal of Algebra, Vol: 322, Issue: 7, Date: October 1, 2009, Pages: 2590-2600

Mathematics

Representation Theory

11 pages

Scientific paper

We study several families of semisimple Hopf algebras, arising as bismash products, which are constructed from finite groups with a certain specified factorization. First we associate a bismash product $H_q$ of dimension $q(q-1)(q+1)$ to each of the finite groups $PGL_2(q)$ and show that these $H_q$ do not have the structure (as algebras) of group algebras (except when $q =2,3$). As a corollary, all Hopf algebras constructed from them by a comultiplication twist also have this property and are thus non-trivial. We also show that bismash products constructed from Frobenius groups do have the structure (as algebras) of group algebras.

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