Invariants of universal enveloping algebras of relatively free Lie algebras

Let F-m(M) be the relatively free algebra of rank m greater than or equal t o 2 in the nonlocally nilpotent variety M of Lie algebras over an infinite field of any characteristic. We study the problem of finite generation of t he 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 (F-m(M)). if the matrix g has eigenvalues of different multiplicative order s, then we show that the algebra of invariants U(F-m(M))(G) is not finitely generated. If all eigenvalues of g ate of the same order and M is a subvar iety of the variety N-c U of all nilpotent of class c-by-abelian algebras f or some c greater than or equal to 1, then the algebra of invariants is fin itely generated. On the other hand, for every g which is not a scalar matri x, there exists a variety of Lie algebras M such that the algebra U(F-m(M)) (G) is not finitely generated. (C) 2000 Academic Press.