Bahturin, Y. A.; Linchenko, V. [Bakhturin, Yu. A.] Identities of algebras with actions of Hopf algebras. (English) Zbl 0912.16009 J. Algebra 202, No. 2, 634-654 (1998). Let \(H\) be a finite-dimensional Hopf algebra acting on an associative algebra \(A\). The paper under review is devoted to the problem whether \(A\) satisfies a non-trivial polynomial identity provided that the subalgebra of invariants \(A^H\) has a non-trivial identity and, if this is the case, to determine whether the degrees of the identities of \(A\) and \(A^H\) are related in terms of the Hopf algebra only. Hopf algebra actions on algebras appear quite often in ring theory. For example, group graded algebras can be considered with the action of a Hopf algebra dual to the group algebra.The main result of the paper under review shows that natural conditions on the Hopf algebra itself are equivalent to the positive answer to the problem under consideration and to other problems of similar type. This implies the equivalence of the following statements if they hold for all associative algebras \(A\) with action of a fixed Hopf algebra \(H\): (1) \(A^H\) is PI implies that \(A\) is PI; (2) There exists a function \(f(t)\) such that if \(A^H\) satisfies an identity of degree \(t\), then \(A\) satisfies an identity of degree bounded by \(f(t)\); (3) \(A^H\) is nilpotent implies that \(A\) is nilpotent; (4) There exists a function \(g(t)\) such that if \(A^H\) is nilpotent of class \(t\), then \(A\) is nilpotent of class bounded by \(g(t)\). Any of these conditions implies that the Hopf algebra \(H\) has to be semisimple. As a consequence, the authors consider both old and recent results of Bergen-Cohen, Kharchenko and Bahturin-Giambruno-Riley from another point of view and give new proofs. Reviewer: Vesselin Drensky (Sofia) Cited in 1 ReviewCited in 24 Documents MSC: 16R10 \(T\)-ideals, identities, varieties of associative rings and algebras 16T05 Hopf algebras and their applications 16R30 Trace rings and invariant theory (associative rings and algebras) 16W20 Automorphisms and endomorphisms 16W50 Graded rings and modules (associative rings and algebras) Keywords:algebras with polynomial identities; finite-dimensional Hopf algebras; invariants of Hopf algebras; graded algebras; algebras of invariants; Hopf algebra actions PDFBibTeX XMLCite \textit{Y. A. Bahturin} and \textit{V. Linchenko}, J. Algebra 202, No. 2, 634--654 (1998; Zbl 0912.16009) Full Text: DOI References: [1] Bahturin, Yuri., Identical Relations in Lie Algebras (1987), VNU Science Press: VNU Science Press Utrecht · Zbl 0414.17008 [2] Yu. Bahturin, A. Giambruno, D. Riley, Group-graded algebras satisfying a polynomial identity, Univ. Palermo, 1996; Yu. Bahturin, A. Giambruno, D. Riley, Group-graded algebras satisfying a polynomial identity, Univ. Palermo, 1996 [3] Yu. A. Bahturin, M. V. Zaicev, Identities of graded algebras, Univ. Palermo, 1996; Yu. A. Bahturin, M. V. Zaicev, Identities of graded algebras, Univ. Palermo, 1996 · Zbl 0920.16011 [4] Bergen, G.; Cohen, M., Actions of commutative Hopf algebras, Bull. London Math. Soc., 18, 159-164 (1986) · Zbl 0563.16003 [5] Bergman, G. M.; Isaacs, I. M., Rings with fixed-point-free group actions, Proc. London Math. Soc., 27, 69-78 (1972) · Zbl 0234.16005 [6] De Vincenco, O., Cocharacters of \(G\), Comm. Algebra, 24, 3293-3310 (1996) · Zbl 0880.16013 [7] Giambruno, A.; Regev, A., Wreath products and P.I. algebras, J. Pure Appl. Algebra, 35, 133-149 (1985) · Zbl 0563.16008 [8] Kharchenko, V., Galois extensions and rings of quotients, Algebra i Logika, 460-484 (1974) [9] Montgomery, S., Fixed Rings of Finite Automorphism Groups of Associative Rings, Lecture Notes in Math. (1980), Springer-Verlag: Springer-Verlag Berlin · Zbl 0449.16001 [10] Montgomery, S., Hopf Algebras and Their Actions on Rings, CBMS (1993), Amer. Math. Soc: Amer. Math. Soc Providence · Zbl 0804.16041 [11] S. Montgomery, S. J. Witherspoon, Irreducible representations of crossed products, J. Pure Appl. Algebra; S. Montgomery, S. J. Witherspoon, Irreducible representations of crossed products, J. Pure Appl. Algebra · Zbl 0932.16039 [12] Razmyslov, Yu. P., Identities of Algebras and Their Representations, Transl. Math. Monographs (1994), Amer. Math. Soc: Amer. Math. Soc Providence · Zbl 0827.17001 [13] Sweedler, Hopf Algebras (1969), Benjamin: Benjamin New York This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.