On growth of Lie algebras, generalized partitions, and analytic functions
The article is a review of results concerning two different types of growth of Lie algebras that lead to some combinatorial problems. The first of them is the growth of finitely generated Lie algebras that leads to a study of generalized partitions. Some results on asymptotics for generalized partitions are given in the paper. A main application is an asymptotical result for the growth of free polynilpotent finitely generated Lie algebras. As a corollary, the author specifies the asymptotic growth of lower central series ranks for free polynilpotent finitely generated groups. The second aspect discussed in the article is a codimensional growth for varieties of Lie algebras. In this connection the author considers some exponential generating functions called complexity functions. Those functions are entire functions of a complex variable. The author computes complexity functions for some varieties. The growth of a complexity function for an arbitrary polynilpotent variety is evaluated. As results the asymptotics of the codimensional growth of a polynilpotent variety of Lie algebras and an upper bound for the growth of an arbitrary nontrivial variety of Lie algebras are obtained. No proofs are included.
scientific article; zbMATH DE number 1453360