The maximal abelian dimension of a Lie algebra, Rentschler's property and Milovanov's conjecture

TL;DR

This paper investigates Rentschler's property in various Lie algebras, demonstrating its implications for Milovanov's conjecture, especially in filiform Lie algebras up to dimension eight, and explores the structure of their Poisson centers.

Contribution

It introduces and studies Rentschler's property across different Lie algebra classes and proves Milovanov's conjecture for specific filiform Lie algebras and low-dimensional cases.

Findings

Rentschler's property holds in several classes of Lie algebras.

Milovanov's conjecture is verified for certain filiform Lie algebras.

Poisson centers are explicitly determined for low-dimensional filiform Lie algebras.

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 Ln, Qn, Rn, Wn and also for all filiform Lie algebras of dimension at most eight. For the latter the Poisson center of these Lie algebras is determined.