Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Jiyoung Han

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 $\mathrm{SL}_d(\mathbb R)$.

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

Abstract

In this concise article, we compute the weight decomposition of with respect to the adjoint representation of , where and demonstrate in detail that comprises two irreducible -invariant subspaces. This can be employed to establish the well-known fact that the identity component of is a maximal connected subgroup of .
Paper Structure (4 sections, 5 theorems, 9 equations, 2 tables)

This paper contains 4 sections, 5 theorems, 9 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Restricted Root Space Decomposition of $\mathfrak{so}(p,q)$
  3. Weight Decomposition of $\mathfrak{sl}_d(\mathbb{R})$
  4. Proof of Main Theorem

Key Result

Theorem 1.1

Let $d=p+q\ge 2$ with $p,q\ge 0$. The subalgebra $\mathfrak{sl}_d(\mathbb{R})$ consists of two $\mathfrak{so}(p,q)$-irreducible subspaces.

Theorems & Definitions (7)

  • Theorem 1.1
  • Corollary 1.2
  • Proposition 2.1
  • Proposition 3.1
  • proof
  • Proposition 4.1
  • proof