## Identities in Algebras with Actions of Hopf Algebras

**Yuri Bahturin **

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.