Invariants of the action of a semisimple Hopf algebra on PI-algebra

We extend classical results in the invariant theory of finite groups to the action of a finite-dimensional Hopf algebra $H$ on an algebra satisfying a polynomial identity. In particular, we prove that an $H$-module algebra $A$ over an algebraically closed field $\mathbf k$ is integral over the subalgebra of invariants, if $H$ is a semisimple and cosemisimple Hopf algebra. We show that if $\operatorname{char}\mathbf k>0$, then the algebra $Z(A)^{H_0}$ is integral over the subalgebra of central invariants $Z(A)^H$, where $Z(A)$ is the center of algebra $A$, $H_0$ is the coradical of $H$. This result allowed to prove that the algebra $A$ is integral over the subalgebra $Z(A)^H$ in some special case. We also construct a counterexample to the integrality of the algebra $A^{H_0}$ over the subalgebra of invariants $A^H$ for a pointed Hopf algebra over a field of non-zero characteristic.