Polynomial identity enveloping algebras

Let L be a Lie (super)-algebra over a field F with the enveloping algebra U(L). In this talk we consider the case when U(L) satisfies a polynomial identity (PI). Latyshev proved that if L is a Lie algebra over a field of characteristic zero then U(L) satisfies a PI if and only if L is abelian. Bahturin extended Latyshev's result to positive characteristic. A polynomial identity is called non-matrix if M_2(F) does not satisfy this identity. We shall mainly focus on non-matrix identities and try to give a characterization of Lie superalgebras when their enveloping algebras satisfy a non-matrix PI. The talk will be aimed at graduate-student level and general algebraist.