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A Frobenius–Schur Theorem for Hopf Algebras


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.

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Linchenko, V., Montgomery, S. A Frobenius–Schur Theorem for Hopf Algebras. Algebras and Representation Theory 3, 347–355 (2000).

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  • Hopf algebras
  • semisimple algebras
  • irreducible representations