Exponential generating functions and complexity of Lie varieties

Suppose that V is a variety of Lie algebras, and let c(n)(V) be the dimensi on of the linear span of all multilinear words on n distinct letters in the free algebra F(V, X) of the variety V. We consider an exponential generati ng function [GRAPHICS] called the complexity function. The complexity function is an entire functi on of a complex variable provided the variety of Lie algebras is nontrivial . In this paper we introduce the notion of complexity for Lie varieties in terms of the growth of complexity functions; also we describe what the comp lexity means for the codimension growth of the variety. Our main goal is to specify the complexity of a product of two Lie varieties in terms of the c omplexities of multiplicands. The main observation here is that C(MV, z) be haves like a composition of three functions C(M, z), exp(z), and C(V, z).