Identities in Group Rings, Enveloping Algebras and Poisson Algebras

Abstract

This is a short survey of works on identical relations in group rings, enveloping algebras, Poisson symmetric algebras and other related algebraic structures. First, the classical work of Passman specified group rings that satisfy nontrivial identical relations. This result was an origin and motivation of close research projects. Second, Latyshev and Bahturin determined Lie algebras such that their universal enveloping algebra satisfies a non-trivial identical relation. Next, Passman and Petrogradsky solved a similar problem in case of restricted enveloping algebras. Third, Farkas started to study identical relations in Poisson algebras. On the other hand, Shestakov proved that the symmetric algebra S(L) of an arbitrary Lie algebra L satisfies the identity {x, {y, z}}≡ 0 if, and only if, L is abelian. Also, Giambruno and Petrogradsky determined when a truncated symmetric Poisson algebra satisfies a non-trivial multilinear Poisson identical relation. We survey further results on existence of identical relations in (truncated) symmetric Poisson algebras of Lie algebras. In particular, we report on recent results on (strong) Lie nilpotency and (strong) solvability of (truncated) symmetric Poisson algebras and related nilpotency classes. Also, we discuss constructions and methods to achieve these results.