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Identities of algebras with actions of Hopf algebras. (English) Zbl 0912.16009

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.

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)
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References:

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