Toward a Jacobson--Morozov theorem for Kac--Moody Lie algebras

TL;DR

This paper extends the Jacobson--Morozov theorem from finite-dimensional semisimple Lie algebras to symmetrizable Kac--Moody algebras, providing proofs for rank two hyperbolic cases and more general cases under certain restrictions.

Contribution

It proposes a generalization of the Jacobson--Morozov theorem to symmetrizable Kac--Moody algebras and offers proofs in specific cases.

Findings

Proof of the generalization for rank two hyperbolic Kac--Moody algebras

Extension of the theorem to arbitrary symmetrizable Kac--Moody algebras with restrictions

Establishment of a correspondence between nilpotent elements and subalgebras in the Kac--Moody setting

Abstract

For a finite-dimensional semisimple Lie algebra $\mathfrak{g}$, the Jacobson--Morozov theorem gives a construction of subalgebras $\mathfrak{sl}_2 \subset \mathfrak{g}$ corresponding to nilpotent elements of $\mathfrak{g}$. In this note, we propose an extension of the Jacobson--Morozov theorem to the symmetrizable Kac--Moody setting and give a proof of this generalization in the case of rank two hyperbolic Kac--Moody algebras. We also give a proof for an arbitrary symmetrizable Kac--Moody algebra under some stronger restrictions.