A Cohomological Proof that Real Representations of Semisimple Lie Algebras Have $\mathbb{Q}$-Forms

TL;DR

This paper offers a cohomological proof that all real semisimple Lie algebras possess rational forms, specifically $ ext{Q}$-forms, and classifies those that are $ ext{R}$-universal, enhancing understanding of their algebraic structures.

Contribution

It provides a concise cohomological proof that every real semisimple Lie algebra has a $ ext{Q}$-form and classifies the $ ext{R}$-universal cases.

Findings

Every real semisimple Lie algebra has an $ ext{R}$-universal $ ext{Q}$-form.

Classification of $ ext{R}$-universal semisimple Lie algebras.

Use of Galois cohomology to simplify proofs in Lie algebra theory.

Abstract

A Lie algebra $\mathfrak{g}_\mathbb{Q}$ over $\mathbb{Q}$ is said to be $\mathbb{R}$-universal if every homomorphism from $\mathfrak{g}_\mathbb{Q}$ to $\mathfrak{gl}(n,\mathbb{R})$ is conjugate to a homomorphism into $\mathfrak{gl}(n,\mathbb{Q})$ (for every $n$). By using Galois cohomology, we provide a short proof of the known fact that every real semisimple Lie algebra has an $\mathbb{R}$-universal $\mathbb{Q}$-form. We also provide a classification of the $\mathbb{R}$-universal Lie algebras that are semisimple.