Co-stability of radicals and its applications to PI-theory

TL;DR

This paper proves that radicals in finite dimensional associative and Lie algebras are stable under certain Hopf algebra coactions and uses this to confirm an analog of Amitsur's conjecture for graded polynomial identities.

Contribution

It establishes the co-stability of radicals under Hopf algebra coactions and applies this to prove an analog of Amitsur's conjecture for graded identities.

Findings

Jacobson radical is an H-subcomodule under specified conditions.

Radicals are graded ideals in group-graded finite dimensional algebras.

Confirmed the analog of Amitsur's conjecture for graded polynomial identities.

Abstract

We prove that if A is a finite dimensional associative H-comodule algebra over a field F for some involutory Hopf algebra H not necessarily finite dimensional, where either char F = 0 or char F > dim A, then the Jacobson radical J(A) is an H-subcomodule of A. In particular, if A is a finite dimensional associative algebra over such a field F, graded by any group, then the Jacobson radical J(A) is a graded ideal of A. Analogous results hold for nilpotent and solvable radicals of finite dimensional Lie algebras over a field of characteristic 0. We use the results obtained to prove the analog of Amitsur's conjecture for graded polynomial identities of finite dimensional associative algebras over a field of characteristic 0, graded by any group.