A geometric characterization of the finitary special linear and unitary Lie algebras

TL;DR

This paper characterizes finitary special linear and unitary Lie algebras through their extremal geometries, revealing structural insights and relationships with classical Lie algebra geometries.

Contribution

It provides a geometric characterization of finitary special linear and unitary Lie algebras based on extremal elements and their associated geometries.

Findings

Finitary special linear Lie algebras are characterized by their extremal geometry.

Finitary special unitary Lie algebras are characterized by geometries with no lines, which become lines after quadratic field extension.

The extremal geometry uniquely determines these Lie algebras.

Abstract

An extremal element $x$ in a Lie algebra $\mathfrak{g}$ is an element for which the space $[x, [x, \mathfrak{g}]]$ is contained in the linear span of $x$. Long root elements in classical Lie algebras are examples of extremal elements. Lie algebras generated by extremal elements lead to geometries with as points the $1$-spaces generate by extremal elements an as lines the $2$-spaces whose non-zero elements are pairwise commuting extremal elements. In this paper we show that the finitary special linear Lie algebras can be characterized by their extremal geometry. Moreover, we also show that the finitary special unitary Lie algebras can be characterized by the fact that their geometry has no lines, but that after extending the field quadratically, the geometry is that of a special linear Lie algebra.