Co-stability of Radicals and Its Applications to PI-Theory

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. In addition, we provide a criterion for graded simplicity of an associative algebra in terms of graded codimensions.