Commutators and Cartan subalgebras in Lie algebras of compact semisimple Lie groups

TL;DR

The paper provides new proofs for key theorems about Lie algebras of compact semisimple Lie groups, demonstrating the existence of orthogonal Cartan subalgebras and exploring their properties and applications.

Contribution

It introduces novel proofs avoiding classification tables, establishes the existence of orthogonal Cartan subalgebras, and extends results to certain non-Hermitian real semisimple Lie algebras.

Findings

Every element in the Lie algebra can be expressed as a commutator.

Orthogonal Cartan subalgebras always exist in compact semisimple Lie algebras.

The commutator map is open at zero in these Lie algebras.

Abstract

First we give a new proof of Goto's theorem for Lie algebras of compact semisimple Lie groups using Coxeter transformations. Namely, every $x$ in $L = \operatorname{Lie}(G)$ can be written as $x =[a, b]$ for some $a$, $b$ in $L$. By using the same method, we give a new proof of the following theorem (thus avoiding the classification tables of fundamental weights): in compact semisimple Lie algebras, orthogonal Cartan subalgebras always exist (where one of them can be chosen arbitrarily). Some of the consequences of this theorem are the following. $(i)$ If $L=\operatorname{Lie}(G)$ is such a Lie algebra and $C$ is any Cartan subalgebra of $L$, then the $G$-orbit of $C^{\perp}$ is all of $L$. $(ii)$ The consequence in part $(i)$ answers a question by L. Florit and W. Ziller on fatness of certain principal bundles. It also shows that in our case, the commutator map $L \times L \to L$ is open at $(0, 0)$. $(iii)$ given any regular element $x$ of $L$, there exists a regular element $y$ such that $L = [x, L] + [y,L]$ and $x$, $y$ are orthogonal. Then we generalize this result about compact semisimple Lie algebras to the class of non-Hermitian real semisimple Lie algebras having full rank. Finally, we survey some recent related results , and construct explicitly orthogonal Cartan subalgebras in $\mathfrak{su}(n)$, $\mathfrak{sp}(n)$, $\mathfrak{so}(n)$.