Weight decomposition of $\mathfrak{sl}_d(\mathbb R)$ with respect to the adjoint representation of $\mathfrak{so}(p,q)$

TL;DR

This paper computes the weight decomposition of the Lie algebra rak{sl}_d(\u211b) under the adjoint action of rak{so}(p,q), revealing its structure and implications for subgroup maximality.

Contribution

It explicitly determines the weight decomposition of rak{sl}_d(\u211b) relative to rak{so}(p,q), showing the algebra splits into two invariant subspaces and clarifies subgroup maximality.

Findings

rak{sl}_d(\u211b) has two rak{so}(p,q)-invariant subspaces.

The decomposition aids in understanding the subgroup structure of rak{so}(p,q) within rak{sl}_d().

The identity component of rak{SO}(p,q) is a maximal connected subgroup of rak{SL}_d().

Abstract

In this concise article, we compute the weight decomposition of $\mathfrak{sl}_d(\mathbb R)$ with respect to the adjoint representation of $\mathfrak{so}(p,q)$, where $d=p+q$ and demonstrate in detail that $\mathfrak{sl}_d(\mathbb R)$ comprises two irreducible $\mathfrak{so}(p,q)$-invariant subspaces. This can be employed to establish the well-known fact that the identity component of $\mathrm{SO}(p,q)$ is a maximal connected subgroup of $\mathrm{SL}_d(\mathbb R)$.