Polynomial identities of algebras with actions of pointed Hopf algebras

Let a pointed Hopf algebra H, over a field , be generated as an algebra by the finite group G=G(H) of group-like elements of H and by a coideal , which satisfies the normalizing condition . If we additionally assume that H is generated by group-like and skew primitive elements. It is proved that if A is a semiprime H-module algebra and acts on A finitely and nilpotently with the semiprime subalgebra of invariants , then A satisfies a polynomial identity if and only if satisfies a polynomial identity. Applications of this result to actions of concrete Hopf algebras on semiprime algebras are described.