Polynomial identities of algebras with actions of pointed Hopf algebras. (Q1882994)

Let \(H\) be a Hopf algebra over a field \(k\), let \(A\) be a \(k\)-algebra, and assume that \(A\) is an \(H\)-module algebra. The paper is devote to the following general problem: if \(H\) is finite dimensional and the subalgebra of invariants \(A^H\) satisfies a polynomial identity (PI), must \(A\) also satisfy a PI? The main result of this paper states that if \(H\) is a Hopf algebra over a field \(k\) generated as an algebra by the finite group \(G\) of group-like elements and by a coideal \(I\) which satisfies the normalizing condition \(Ik[G]=k[G]I\) (if \(\text{char\,}k=0\) it is assumed that \(H\) is generated by group-like elements and skew primitive elements), \(A\) is a semiprime \(H\)-module algebra and \(I\) acts on \(A\) finitely and nilpotently with the semiprime subalgebra of invariants \(A^H\), then \(A\) satisfies a polynomial identity if and only if \(A^H\) satisfies a polynomial identity. In the last section some examples and applications are presented.