A geometric characterization of the symplectic Lie algebra

TL;DR

This paper provides a geometric characterization of finitary symplectic Lie algebras by analyzing extremal elements and their algebraic relations, offering a new perspective on their structure.

Contribution

It introduces a novel geometric characterization of symplectic Lie algebras based on extremal elements and their generating properties.

Findings

Finitary symplectic Lie algebras are generated by extremal elements.

Extremal elements satisfy specific commutation relations involving $rak{sl}_2$.

The characterization links algebraic properties to geometric structures.

Abstract

A nonzero element $x$ in a Lie algebra $\mathfrak{g}$ with Lie product $[ , ]$ is called extremal if $[x,[x,y]]$ is a multiple of $x$ for all $y$. In this paper we characterize the (finitary) symplectic Lie algebras as simple Lie algebras generated by their extremal elements satisying the condition that any two noncommuting extremal elements $x,y$ generate an $\mathfrak{sl}_2$ and any third extremal element $z$ commutes with at least one extremal element in this $\mathfrak{sl}_2$.