Actions of Ore extensions and growth of polynomialH-identities

Abstract

Abstract. We show that if A is a finite dimensional associative H-module algebra for an arbitrary Hopf algebra H, then the proof of the analog of Amitsur's conjecture for Hcodimensions of A can be reduced to the case when A is H-simple. (Here we do not require that the Jacobson radical of A is an H-submodule.) As an application, we prove that if A is a finite dimensional associative H-module algebra where H is a Hopf algebra H over a field of characteristic 0 such that H is constructed by an iterated Ore extension of a finite dimensional semisimple Hopf algebra by skew-primitive elements (e.g. H is a Taft algebra), then there exists integer PIexp H (A). In order to prove this, we study the structure of algebras simple with respect to an action of an Ore extension.

Equivalences of (co)module algebra structures over Hopf algebras

“…We begin with a quick introduction into the theory of polynomial H-identities (see also [7,9,20]). All algebras in this section are associative, but not necessarily unital.…”

“…Karasik [25] for the case when A is a not necessarily finite dimensional PI-algebra) and in [16,Theorem 1] for H-module algebras A such that the Jacobson radical J(A) is an H-submodule (the requirement that A/J(A) is a direct sum of H-simple algebras is satisfied by [32, Theorem 1.1], [33,Lemma 4.2]). In the case when J(A) is not an H-submodule, the conjecture was proved only for Hopf algebras H that are iterated Ore extensions of finite dimensional semisimple Hopf algebras [20,Corollary 7.4].…”