dorsal/arxiv
View SchemaSome Properties of Finite-Dimensional Semisimple Hopf Algebras
| Authors | Pavel Etingof, Shlomo Gelaki |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9712033 |
| URL | https://arxiv.org/abs/q-alg/9712033 |
Abstract
Kaplansky conjectured that if H is a finite-dimensional semisimple Hopf algebra over an algebraically closed field k of characteristic 0, then H is of Frobenius type (i.e. if V is an irreducible representation of H then dimV divides dimH). It was proved by Montgomery and Witherspoon that the conjecture is true for H of dimension p^n, p prime, and by Nichols and Richmond that if H has a 2-dimensional representation then dimH is even. In this paper we first prove that if V is an irreducible representation of D(H), the Drinfeld double of any finite-dimensional semisimple Hopf algebra H over k, then dimV divides dimH (not just dimD(H)=(dimH)^2). In doing this we use the theory of modular tensor categories (in particular Verlinde formula). We then use this statement to prove that Kaplansky's conjecture is true for finite-dimensional semisimple quasitriangular Hopf algebras over k. As a result we prove easily the result of Zhu that Kaplansky's conjecture on prime dimensional Hopf algebras over k is true, by passing to their Drinfeld doubles. Second, we use a theorem of Deligne on characterization of tannakian categories to prove that triangular semisimple Hopf algebras over k are equivalent to group algebras as quasi-Hopf algebras.
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"abstract": "Kaplansky conjectured that if H is a finite-dimensional semisimple Hopf\nalgebra over an algebraically closed field k of characteristic 0, then H is of\nFrobenius type (i.e. if V is an irreducible representation of H then dimV\ndivides dimH). It was proved by Montgomery and Witherspoon that the conjecture\nis true for H of dimension p^n, p prime, and by Nichols and Richmond that if H\nhas a 2-dimensional representation then dimH is even. In this paper we first\nprove that if V is an irreducible representation of D(H), the Drinfeld double\nof any finite-dimensional semisimple Hopf algebra H over k, then dimV divides\ndimH (not just dimD(H)=(dimH)^2). In doing this we use the theory of modular\ntensor categories (in particular Verlinde formula). We then use this statement\nto prove that Kaplansky\u0027s conjecture is true for finite-dimensional semisimple\nquasitriangular Hopf algebras over k. As a result we prove easily the result of\nZhu that Kaplansky\u0027s conjecture on prime dimensional Hopf algebras over k is\ntrue, by passing to their Drinfeld doubles. Second, we use a theorem of Deligne\non characterization of tannakian categories to prove that triangular semisimple\nHopf algebras over k are equivalent to group algebras as quasi-Hopf algebras.",
"arxiv_id": "q-alg/9712033",
"authors": [
"Pavel Etingof",
"Shlomo Gelaki"
],
"categories": [
"q-alg",
"math.QA"
],
"title": "Some Properties of Finite-Dimensional Semisimple Hopf Algebras",
"url": "https://arxiv.org/abs/q-alg/9712033"
},
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