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

On the Siegel-Weil Theorem for Loop Groups (I)

We proved a new Siegel-Weil formula for orthogonal and symplectic groups, which will be used later to prove a generalization of Siegel-Weil formula for loop groups

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

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

A Construction of Representations of Loop Group and Affine Lie Algebra of sln\mathfrak{sl}_n

In this paper, we construct a representation of loop group and derive the formula of the corresponding representation of the affine Kac-Moody algebra with level 1. And we also provide a concrete realization of Whittaker functionals in the dual representation.Comment: 20 pages. Comments are welcome