Characters and a Verlinde-type formula for symmetric Hopf algebras |
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Authors: | Miriam Cohen Sara Westreich |
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Affiliation: | aDepartment of Mathematics, Ben Gurion University of the Negev, Beer Sheva, Israel;bInterdisciplinary Department of the Social Sciences, Bar-Ilan University, Ramat-Gan, Israel |
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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 ΛH, where Λ is an integral of H and is the left adjoint action of H on itself. We describe ΛH 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 ΛH 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 ΛH. |
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Keywords: | Symmetric algebras Characters Unimodular Hopf algebras Factorizable ribbon Hopf algebras |
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