Subrings of invariants for actions of finite-dimensional Hopf algebras (Q2036472)

The initial problem of invariant theory, dealing with a finite group acting on a polynomial ring, has been extended to various contexts (groups acting on a noncommutative ring, Lie algebras acting on a ring\(\ldots\)). All these have been unified in the 80s with Hopf algebra actions, but the level of knowledge in this setting is not so high as in the case of group actions. This paper is a review of recent results on this topic, due to Etingof, Eryashkin, Bergen, Cohen, Fischman and Skryabin among others. An accent is put on the integrality of an algebra with polynomial identities over its subalgebra of invariants over the action of a Hopf algebra.