Actions of Ore extensions and growth of polynomial H-identities

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 H-codimensions 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.

KEYWORDS:

• Associative algebra
• codimension
• H-module algebra
• H-simple algebra
• Ore extension
• polynomial identity
• skew-derivation
• Taft algebra

2010 Mathematics Subject Classification:

• Primary 16R10
• Secondary 16R50, 16T05, 16W25