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Invariants of the action of a semisimple Hopf algebra on PI-algebra

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Abstract

We extend several classical results in the theory of invariants of finite groups to the case of action of a finite-dimensional Hopf algebra H on an algebra satisfying a polynomial identity. In particular, we prove that an H-module algebra A over an algebraically closed field k is integral over the subalgebra of invariants, if H is a semisimple and cosemisimple Hopf algebra. We show that for char k > 0, the algebra Z \({\left( A \right)^{{H_0}}}\) is integral over the subalgebra of central invariants Z(A)H, where Z(A) is the center of algebra A, H 0 is the coradical of H. This result allowed us to prove that the algebra A is integral over the subalgebra Z(A)H in some special case. We also construct a counterexample to the integrality of the algebra \({A^{{H_0}}}\) over the subalgebra of invariants A H for a pointed Hopf algebra over a field of non-zero characteristic.

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Authors and Affiliations

  1. Kazan (Volga Region) Federal University, ul. Kremlyovskaya 18, Kazan, 420008, Russia

    M. S. Eryashkin

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  1. M. S. Eryashkin
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Correspondence to M. S. Eryashkin.

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Original Russian Text © M.S. Eryashkin, 2016, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2016, No. 8, pp. 21–34.

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Eryashkin, M.S. Invariants of the action of a semisimple Hopf algebra on PI-algebra. Russ Math. 60, 17–28 (2016). https://doi.org/10.3103/S1066369X1608003X

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