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A Frobenius–Schur Theorem for Hopf Algebras

Abstract

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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References

  1. Etingof, P. and Gelaki, S.: On finite-dimensional semisimple and cosemisimple Hopf algebras in positive characteristic, Inter. Math. Res. Notices 16 (1998), 851–864.

    Google Scholar 

  2. Etingof, P. and Gelaki, S.: On the exponent of finite-dimensional Hopf algebras, Math Res. Lett. 6 (1999), 131–140.

    Google Scholar 

  3. Martin Isaacs, I.: Character Theory of Finite Groups, Dover, New York, 1994.

    Google Scholar 

  4. Jacobson, N.: Finite-Dimensional Division Algebras, Springer-Verlag, Berlin, 1996.

    Google Scholar 

  5. Kashina, Y.: On the order of the antipode of Hopf algebras in H H YD, Comm. Algebra 27 (1999), 1261–1273.

    Google Scholar 

  6. Kashina, Y.: The classification of semisimple Hopf algebras of dimension 16, J. Algebra 232 (2000), 617–663.

    Google Scholar 

  7. Larson, R.: Characters of Hopf algebras, J. Algebra 17 (1971), 352–368.

    Google Scholar 

  8. Larson, R. and Radford, D.: Semisimple cosemisimple Hopf algebras, Amer. J. Math. 110 (1988), 187–195.

    Google Scholar 

  9. Montgomery, S.: Hopf Algebras and Their Actions on Rings, CBMS 82, Amer.Math. Soc., Providence, 1993.

  10. Nikshych, D.: Corrigendum: K 0-rings and twisting of finite-dimensional semisimple Hopf algebras, Comm. Algebra 26 (1998), 2019.

    Google Scholar 

  11. Oberst, U. and Schneider, H.-J.: Ñber Untergruppen endlicher algebraischer Gruppen, Manuscripta Math. 8 (1973), 217–241.

    Google Scholar 

  12. Schneider, H.-J.: Lectures on Hopf algebras, Universidad de Cordoba Trabajos de Matematica, No. 31/95, Cordoba (Argentina), 1995.

  13. Serre, J. P.: Linear Representations of Finite Groups, Springer-Verlag, Berlin and New York, 1977.

    Google Scholar 

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Linchenko, V., Montgomery, S. A Frobenius–Schur Theorem for Hopf Algebras. Algebras and Representation Theory 3, 347–355 (2000). https://doi.org/10.1023/A:1009949909889

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  • DOI: https://doi.org/10.1023/A:1009949909889

  • Hopf algebras
  • semisimple algebras
  • irreducible representations