Moscow State University
We first explain how the actions of Hopf algebras naturally arise when people consider graded algebras, automorphism groups and derivation algebras. Then we discuss the following problem. Let A be an algebra with an action of a finte-dimensional Hopf algebra H, and AH the subalgebra of invariants of the action. Suppose AH satisfies a non-trivial identity (say, commutative or nilpotent). For what H can one conclude that also A satisfies a non-trivial identity? Among several results to be mentioned we formulate the following solution of a problem due to A. Zalesskii. Let L be a Lie algebra graded by a finite group G and L1 satisfies a non-trivial identity of degree d. Then L satisfies such an identity of degree f(d,|G|) depending only on d and |G| but not on L itself.