The Maximal Abelian Dimension of a Lie Algebra, Rentschler's Property and Milovanov's Conjecture

Abstract

A finite dimensional Lie algebra L with magic number c(L) is said to satisfy Rentschler's property if it admits an abelian Lie subalgebra H of dimension at least c(L) − 1. We study the occurrence of this new property in various Lie algebras, such as nonsolvable, solvable, nilpotent, metabelian and filiform Lie algebras. Under some mild condition H gives rise to a complete Poisson commutative subalgebra of the symmetric algebra S(L). Using this, we show that Milovanov's conjecture holds for the filiform Lie algebras of type L n , Q n , R n , W n and also for all filiform Lie algebras of dimension at most eight. For the latter the Poisson center of these Lie algebras is determined.