Hopf Algebras and Congruence Subgroups

We prove that the kernel of the natural action of the modular group on the center of the Drinfel'd double of a semisimple Hopf algebra is a congruence subgroup. To do this, we introduce a class of generalized Frobenius-Schur indicators and endow it with an action of the modular group that is compatible with the original one.Comment: 130 pages. Many new results added, remark by D. Nikshych included. See also http://www.southalabama.edu/mathstat/personal_pages/sommerh

Self-dual modules of semisimple Hopf algebras

We prove that, over an algebraically closed field of characteristic zero, a semisimple Hopf algebra that has a nontrivial self-dual simple module must have even dimension. This generalizes a classical result of W. Burnside. As an application, we show under the same assumptions that a semisimple Hopf algebra that has a simple module of even dimension must itself have even dimension.Comment: 9 pages. Important new result included. See also http://www.mathematik.uni-muenchen.de/~sommer

On Higher Frobenius-Schur Indicators

We study the higher Frobenius-Schur indicators of modules over semisimple Hopf algebras, and relate them to other invariants as the exponent, the order, and the index. We prove various divisibility and integrality results for these invariants. In particular, we prove a version of Cauchy's theorem for semisimple Hopf algebras. Furthermore, we give some examples that illustrate the general theory.Comment: 62 pages. Important new result added, remark by P. Etingof included, mistake in last section corrected. See also http://www.mathematik.uni-muenchen.de/~sommer

On Isomorphisms between Certain Yetter-Drinfel'd Hopf Algebras

For two families of Yetter-Drinfel'd Hopf algebras considered earlier by the authors, we determine which of them are isomorphic. We also determine which of their Radford biproducts are isomorphic.Comment: 23 pages. In the second version, the argument in Paragraph 1.5 has been simplified. See also https://www.math.mun.ca/~sommerh

Stable anti-Yetter-Drinfeld modules

We define and study a class of entwined modules (stable anti-Yetter-Drinfeld modules) that serve as coefficients for the Hopf-cyclic homology and cohomology. In particular, we explain their relationship with Yetter-Drinfeld modules and Drinfeld doubles. Among sources of examples of stable anti-Yetter-Drinfeld modules, we find Hopf-Galois extensions with a flipped version of the Miyashita-Ulbrich action