Root Fernando-Kac subalgebras of finite type

TL;DR

This paper proves a conjecture characterizing Fernando-Kac subalgebras of finite type containing a Cartan subalgebra in simple Lie algebras, excluding E8, using combinatorial criteria.

Contribution

It confirms Penkov's conjecture for all simple Lie algebras except E8, providing a clear combinatorial description of Fernando-Kac subalgebras of finite type.

Findings

Confirmed Penkov's conjecture for all simple Lie algebras except E8.

Provided explicit combinatorial criteria for Fernando-Kac subalgebras of finite type.

Enhanced understanding of the structure of subalgebras related to module actions.

Abstract

Let $\mathfrak{g}$ be a finite-dimensional Lie algebra and $M$ be a $\mathfrak{g}$-module. The Fernando-Kac subalgebra of $\mathfrak{g}$ associated to $M$ is the subset $\mathfrak{g}[M]\subset\mathfrak{g}$ of all elements $g\in\mathfrak{g}$ which act locally finitely on $M$. A subalgebra $\mathfrak{l}\subset\mathfrak{g}$ for which there exists an irreducible module $M$ with $\mathfrak{g}[M]=\mathfrak{l}$ is called a Fernando-Kac subalgebra of $\mathfrak{g}$. A Fernando-Kac subalgebra of $\mathfrak{g}$ is of finite type if in addition $M$ can be chosen to have finite Jordan-H\"older $\mathfrak{l}$-multiplicities. Under the assumption that $\mathfrak{g}$ is simple, I. Penkov has conjectured an explicit combinatorial criterion describing all Fernando-Kac subalgebras of finite type which contain a Cartan subalgebra. In the present paper we prove this conjecture for $\mathfrak{g}\neq E_8$.