On types of overexponential growth in Lie PI-algebras

The growth function of identities $c_n(\mathcal{V})$ for varieties of Lie algebras is studied; where $c_n(\mathcal{V})$ is the dimension of a linear span of multilinear words in $n$ distinct letters in a free algebra $F(\mathcal{V},X)$ of the variety $\mathcal{V}$. The main results are as follows: the description of types of overexponential growth is suggested; the growth of identities for polynilpotent varieties is found. A complexity function $\mathcal{C}(\mathcal{V},z)$ is used; it corresponds to any nontrivial variety of Lie algebras $\mathcal{V}$ and is an entire function of a complex variable.