Characters and a Verlinde-type formula for symmetric Hopf algebras

Abstract

We study certain aspects of finite-dimensional non-semisimple symmetric Hopf algebras H and their duals H ∗ . We focus on the set I ( H ) of characters of projective H-modules which is an ideal of the algebra of cocommutative elements of H ∗ . This ideal corresponds via a symmetrizing form to the projective center (Higman ideal) of H which turns out to be Λ ▪ H , where Λ is an integral of H and ▪ is the left adjoint action of H on itself. We describe Λ ▪ H via primitive and central primitive idempotents of H. We also show that it is stable under the quantum Fourier transform. Our best results are obtained when H is a factorizable ribbon Hopf algebra over an algebraically closed field of characteristic 0. In this case Λ ▪ H is also the image of I ( H ) under a “translated” Drinfel'd map. We use this fact to prove the existence of a Steinberg-like character. The above ingredients are used to prove a Verlinde-type formula for Λ ▪ H .