Invariants of the action of a semisimple finite-dimensional Hopf algebra on special algebras

In this paper we extend classical results of the invariant theory of finite groups to the action of a finite-dimensional semisimple Hopf algebra $H$ on a special algebra $A$, which is homomorphically mapped onto a commutative integral domain, and the kernel of this map contains no nonzero $H$-stable ideal. We prove that the algebra $A$ is finitely generated as a module over a subalgebra of invariants, and the latter is finitely generated as a $\mathbf k$-algebra. We give a counterexample for the finite generation of a non-semisimple Hopf algebra.