Higher genus mapping class group invariants from factorizable Hopf algebras

Factorizable quasi-Hopf algebras. Applications

We define the notion of factorizable quasi-Hopf algebra by using a categorical point of view. We show that the Drinfeld double D(H) of any finite dimensional quasi-Hopf algebra H is factorizable, and we characterize D(H) when H itself is factorizable. Finally, we prove that any finite dimensional factorizable quasi-Hopf algebra is unimodular. In particular, we obtain that the Drinfeld double D(...

Invariants of Hopf Algebras

In the winter of 1999 I gave a series of lectures at Queen’s university about some recent results concerning the Cohen-Macaulay property of invariants of Hopf algebras. Tony Geramita asked me to write up my notes for the Queen’s Papers, and I happily took up his suggestion. Although this article focuses on the proof of one main theorem (Theorem 2.11 on page 12), it has some of the character of ...

Central Invariants and Higher Indicators for Semisimple Quasi-hopf Algebras

In this paper, we define the higher Frobenius-Schur (FS-)indicators for finite-dimensional modules of a semisimple quasi-Hopf algebra H via the categorical counterpart developed in a 2005 preprint. When H is an ordinary Hopf algebra, we show that our definition coincides with that introduced by Kashina, Sommerhäuser, and Zhu. We find a sequence of gauge invariant central elements of H such that...

Invariants of finite Hopf algebras

This paper extends classical results in the invariant theory of finite groups and finite group schemes to the actions of finite Hopf algebras on commutative rings. Suppose that H is a finite dimensional Hopf algebra and A a commutative algebra, say over a field K. Let δ : A → A ⊗H be an algebra homomorphism which makes A into a right H-comodule. In this case A is called an H-comodule algebra. T...

The Mapping Class Group of a Genus