Semisimple Hopf algebras: Their representations and actions

University Of Southern California, Los Angeles CA

Investigators

Abstract

The main part of this project concerns certain invariants of the representations of semisimple Hopf algebras. These invariants, the Frobenius-Schur indicators and their higher analogs, should give information not only about the representations of a Hopf algebra, but also about its possible structure and so may help in the classification program. They should also serve as a tool in proving more general facts about Hopf algebras, by looking for representation-thoeretic proofs. The project also concerns actions of Hopf algebras on non-commutative algebras, especially as to when the Jacobson or prime radicals are stable under such actions; these questions are related to prime ideals and the radical of semidirect products, and would help in doing ``invariant theory'' for Hopf algebras. Finally some questions will be considered about twistings of Hopf algebras and of their actions on algebras, as well as more general questions about which algebra properties are preserved by twisting. Hopf algebras are of interest not only for their own sake, but because they appear as invariants in other mathematical structures, such as understanding knots in geometry, and in parts of mathematical physics, such as conformal field theory. Thus a greater understanding of the Hopf algebras themselves might eventually be useful in these areas. This project will also involve some women students involved with the Women in Science and Engineering (WiSE) program at USC.