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Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2016, Number 8, Pages 21–34
(Mi ivm9140)
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This article is cited in 2 scientific papers (total in 2 papers)
Invariants of the action of a semisimple Hopf algebra on PI-algebra
M. S. Eryashkin Kazan (Volga Region) Federal University, 18 Kremlyovskaya str., Kazan, 420008 Russia
Abstract:
We extend classical results in the invariant theory of finite groups to the 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 $\mathbf k$ is integral over the subalgebra of invariants, if $H$ is a semisimple and cosemisimple Hopf algebra. We show that if $\operatorname{char}\mathbf k>0$, then the algebra $Z(A)^{H_0}$ 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 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.
Keywords:
Hopf algebras, invariant theory, PI-algebras, rings of quotients, coradical.
Received: 25.12.2014
Citation:
M. S. Eryashkin, “Invariants of the action of a semisimple Hopf algebra on PI-algebra”, Izv. Vyssh. Uchebn. Zaved. Mat., 2016, no. 8, 21–34; Russian Math. (Iz. VUZ), 60:8 (2016), 17–28
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https://www.mathnet.ru/eng/ivm9140 https://www.mathnet.ru/eng/ivm/y2016/i8/p21
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Abstract page: | 156 | Full-text PDF : | 35 | References: | 27 | First page: | 1 |
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