Root Systems for Levi Factors and Borel-de Siebenthal Theory

TL;DR

This paper develops a theory of $rak{t}$-roots for Levi factors in complex semisimple Lie algebras, extending classical root theory and applying it to structure theorems for nilradicals and Borel-de Siebenthal subalgebras.

Contribution

It introduces a new $rak{t}$-root theory that generalizes classical roots and applies it to analyze nilradicals and irreducibility in Borel-de Siebenthal theory.

Findings

Established a $rak{t}$-root structure analogous to classical root systems.

Derived new structural results for the nilradical $rak{n}$ of $rak{q}.

Proved irreducibility theorems for certain subalgebras using $rak{t}$-roots.

Abstract

Let $\frak{m}$ be a Levi factor of a proper parabolic subalgebra $\frak{q}$ of a complex semisimple Lie algebra $\frak{g}$. Let $\frak{t} = cent \frak{m}$. A nonzero element $\nu \in \frak{t}^*$ is called a $\frak {t}$-root if the corresponding adjoint weight space $\frak{g}_{nu}$ is not zero. If $\nu$ is a $\frak{t}$-root, some time ago we proved that $\frak{g}_{\nu}$ is $ad \frak{m}$ irreducible. Based on this result we develop in the present paper a theory of $\frak{t}$-roots which replicates much of the structure of classical root theory (case where $\frak{t}$ is a Cartan subalgebra). The results are applied to obtain new reults about the structure of the nilradical $\frak{n}$ of $\frak{q}$. Also applications in the case where $dim \frak{t}=1$ are used in Borel-de Siebenthal theory to determine irreducibility theorems for certain equal rank subalgebras of $\frak{g}$. In fact the irreducibility results readily yield a proof of the main assertions of the Borel-de Siebenthal theory.