GROWTH OF SOLVABLE LIE SUPERALGEBRAS#

Abstract

ABSTRACT Finitely generated solvable Lie algebras have an intermediate growth between polynomial and exponent. Recently, the second author suggested the scale to measure such an intermediate growth of Lie algebras. The growth was specified for solvable Lie algebras F(A q , k), the latter being relatively free algebras with k generators and fixed solubility length q. Later, an application of generating functions allowed us to obtain a more precise asymptotic. These results were obtained in the generality of polynilpotent Lie algebras. Now we consider the case of Lie superalgebras. Our goal is to compute the growth for F(A q , m, k), the free solvable Lie superalgebra of length q with m even and k odd generators. The proof is based upon a precise formula of the generating function for this algebra obtained earlier. The result is proved in generality of free polynilpotent Lie superalgebras. We study the growth for universal enveloping algebras of Lie superalgebras as well. Also, we study bases for free Lie superalgebras.