Subalgebra generated by ad-locally nilpotent elements of Borcherds Generalized Kac-Moody Lie algebras

TL;DR

This paper characterizes the subalgebra generated by ad-locally nilpotent elements in Borcherds generalized Kac-Moody Lie algebras, showing it aligns with the Levi subalgebra associated with real simple roots.

Contribution

It identifies the subalgebra generated by ad-locally nilpotent elements as essentially the Levi subalgebra linked to real simple roots in Borcherds GKM Lie algebras.

Findings

The subalgebra generated by ad-locally nilpotent elements is characterized.

This subalgebra coincides with the Levi subalgebra associated with real simple roots.

The structure of the subalgebra is explicitly described in relation to the root system.

Abstract

We determine the Lie subalgebra $\mathfrak{g}_{nil}$ of a Borcherds symmetrizable generalized Kac-Moody Lie algebra $\mathfrak{g}$ generated by $ad$-locally nilpotent elements and show that it is `essentially' the same as the Levi subalgebra of $\mathfrak{g}$ with its simple roots precisely the real simple roots of $\mathfrak{g}$.