Asymptotics of H-identities for associative algebras with an H-invariant radical

Abstract

We prove the existence of the Hopf PI-exponent for finite dimensional associative algebras $A$ with a generalized Hopf action of an associative algebra $H$ with $1$ over an algebraically closed field of characteristic $0$ assuming only the invariance of the Jacobson radical $J(A)$ under the $H$-action and the existence of the decomposition of $A/J(A)$ into the sum of $H$-simple algebras. As a consequence, we show that the analog of Amitsur's conjecture holds for $G$-codimensions of finite dimensional associative algebras over a field of characteristic $0$ with an action of an arbitrary group $G$ by automorphisms and anti-automorphisms and for differential codimensions of finite dimensional associative algebras with an action of an arbitrary Lie algebra by derivations.

arXiv comment9 pages; this article is a generalization of arXiv:1203.5384. In particular, Lemma 3 is a modification of Lemma 10 in arXiv:1203.5384 by the trick from arXiv:1211.1272. The exposition is parallel to that in Section 5 of arXiv:1211.1272. Definitions in Section 2 coincide with those in Section 3.1 of arXiv:1203.5384