Kemer's Theory for H-Module Algebras with Application to the PI Exponent

TL;DR

This paper extends Kemer's theory to H-module algebras over a field of zero characteristic, proving key theorems on representability, the Specht problem, and the integrality of the PI exponent.

Contribution

It establishes foundational theorems for H-module algebras, including representability, the Specht problem, and confirms Amitsur's conjecture in this context.

Findings

Every H-module algebra satisfying an ordinary PI has the same identities as a finite dimensional Grassmann envelope.

The Specht problem is solved for H-module PI algebras, showing finite generation of H-T-ideals.

The PI exponent of an H-module algebra is always an integer.

Abstract

Let H be a semisimple finite dimensional Hopf algebra over a field F of zero characteristic. We prove three major theorems: 1. The Representability theorem which states that every H-module (associative) F-algebra W satisfying an ordinary PI, has the same H-identities as the Grassmann envelope of an $H\otimes\left(F\mathbb{Z}/2\mathbb{Z}\right)^{*}$-module algebra which is finite dimensional over a field extension of F. 2. The Specht problem for H-module (ordinary) PI algebras. That is, every H-T-ideal $\Gamma$ which contains an ordinary PI contains H-polynomials $f_{1},...,f_{s}$ which generates $\Gamma$ as an H-T-ideal. 3. Amitsur's conjecture for H-module algebras, saying that the exponent of the H-codimension sequence of an ordinary PI H-module algebra is an integer.