Frobenius divisibility for Hopf algebras

Abstract

ABSTRACT. We present a unified ring theoretic approach, based on properties of the Casimir element of a symmetric algebra, to a variety of known divisibility results for the degrees of irreducible representations of semisimple Hopf algebras in characteristic 0. All these results are motivated by a classical theorem of Frobenius on the degrees of irreducible complex representations of finite groups.

Nearly Frobenius dimension of Frobenius algebras

“…If A is Frobenius then A ∼ = A * and we can consider an isomorphism ψ : A ⊗ A → End k (A). The Casimir element (see [15]) for (A, λ) be a symmetric algebra is defined as the element C A ∈ A ⊗ A that corresponds to Id A ∈ End k (A) under the isomorphism End k (A) ∼ = A ⊗ A. If e i and e i are dual basis with respect to λ, the Casimir element is:…”