A Frobenius–Schur Theorem for Hopf Algebras

By V. Linchenko, Sue Montgomery. 2000

112 citations recorded by the source corpus.

We prove a version of the Frobenius–Schur theorem for a finite-dimensional semisimple Hopf algebra H over an algebraically closed field; if the field has characteristic p not 0, H is also assumed to be cosemisimple. Then for each irreducible representation V of H, we define a Schur indicator for V, which reduces to the classical Schur indicator when H is the group algebra of a finite group. We prove that this indicator is 0 if and only if V is not self-dual. If V is self dual, then the indicator is positive (respectively, negative) if and only if V admits a nondegenerate bilinear symmetric (resp., skew-symmetric) H-invariant form. A more general result is proved for algebras with involution.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/f66377f6037ae059148d11303598dab8e998332a

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Citations in the corpus

On higher Frobenius-Schur indicators (2003, methodology): “In the sequel, two special cases will be of particular importance for us. First, for sequences of length 1, we obviously have h [n] = h(1) � h(2) � . . . � h(n), which establishes the connection with the notation used in [ 27 ].”
Computing the Frobenius–Schur Indicator for Abelian Extensions of Hopf Algebras☆☆☆ (2001, background): “Theorem 1.1. [ LM ] Let H be a semisimple Hopf algebra over an algebraically closed field k. If k has characteristic p 6= 0, assume in addition that p 6= 2 and that H ∗ is semisimple.”
On the Frobenius–Schur indicators for quasi-Hopf algebras (2003, unknown): “Q A ] 7 S ep 2”
Central invariants and higher indicators for semisimple quasi-Hopf algebras (2005, background): “The notion of (degree 2) Frobenius-Schur indicator ν2(V ) of an irreducible representation V of a finite group G has been generalized to simple modules of semisimple Hopf algebras by Linchenko and Montgomery [ LM00 ], to certain C ∗-fusion cate-”