Invariants of universal enveloping algebras of relatively free lie algebras

Let FmW be the relatively free algebra of rank m ≥ 2 in the nonlocally nilpotent variety W of Lie algebras over an infinite field of any characteristic. We study the problem of finite generation of the algebra of invariants of a cyclic linear group G = g of finite order invertible in the base field, acting on the universal enveloping algebra U(FmW). If the matrix g has eigenvalues of different multiplicative orders, then we show that the algebra of invariants U(FmW)G is not finitely generated. If all eigenvalues of g ate of the same order and W is a subvariety of the variety RcU of all nilpotent of class c-by-abelian algebras for some c ≥ 1, then the algebra of invariants is finitely generated. On the other hand, for every g which is not a scalar matrix, there exists a variety of Lie algebras W such that the algebra U(FmW)G is not finitely generated.