On growth of Lie algebras, generalized partitions, and analytic functions

In this paper we discuss some recent results on two diierent types of growth of Lie algebras, that lead to some combinatorial problems. First, we study the growth of nitely generated Lie algebras (sections 1{4). This problem leads to a study of generalized partitions. Recently the author has suggested a series of q-dimensions of algebras Dim q , q 2 N which includes, as rst terms, dimensions of vector spaces, Gelfand-Kirillov dimensions, and superdimensions. These dimensions enabled us to describe the change of a growth in transition from a Lie algebra to its universal enveloping algebra. In fact, this is a result on some generalized partitions. In this paper we give some results on asymptotics for those generalized partitions. As a main application, we obtain an asymptotical result for the growth of free polynilpotent nitely generated Lie algebras. As a corollary, we specify the asymptotic growth of lower central series ranks for free polynilpotent nitely generated groups. We essentially use Hilbert-Poincar e series and some facts on growth of complex functions which are analytic in the unit circle. By growth of such functions we mean their growth when the variable tends to 1. Also we discuss for all levels q = 2; 3 : : : what numbers > 0 can be a q-dimension of some Lie (associative) algebra. Second, we discuss a \codimension growth" for varieties of Lie algebras (sections 5, 6). It is useful to consider some exponential generating functions called complexity functions. Those functions are entire functions of a complex variable provided the varieties of Lie algebras are nontrivial. We compute the complexity functions for some varieties. The growth of a complexity function for an arbitrary polynilpotent variety is evaluated. Here we need to study the connection between the growth of a fast increasing entire function and the behaviour of its Taylor coeecients. As a result we obtain a result for the asymptotics of the codimension growth of a polynilpotent variety of Lie algebras. Also we obtain an upper bound for a growth of an arbitrary nontrivial variety of Lie algebras. R esum e. Dans ce travail nous traitons quelques r esultats r ecents portant sur deux types dii erents de croissance pour les alg ebres de Lie, qui nous conduisent a des probl emes combi-natoires.