Limits of Jordan Lie subalgebras

TL;DR

This paper investigates the relationship between abelian ideals in Borel subalgebras of simple Lie algebras and Jordan Lie subalgebras, showing that abelian ideals are limits of Jordan subalgebras and that the span of all such abelian subalgebras is generated by Jordan subalgebras.

Contribution

It establishes that n-dimensional abelian ideals are limits of Jordan Lie subalgebras and that their span coincides with the span of all n-dimensional abelian Lie subalgebras in g.

Findings

n-dimensional abelian ideals are limits of Jordan Lie subalgebras

The g-module spanned by all n-dimensional abelian Lie subalgebras is generated by Jordan Lie subalgebras

Connection between abelian ideals and Jordan subalgebras in simple Lie algebras

Abstract

Let g be a simple Lie algebra of rank n over C. We show that the n-dimensional abelian ideals of a Borel subalgebra of g are limits of Jordan Lie subalgebras. Combining this with a classical result by Kostant, we show that the g-module spanned by all n-dimensional abelian Lie subalgebras of g is actually spanned by the Jordan Lie subalgebras.