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

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.

References in corpus (1)

• Real representations of semisimple Lie algebras have Q-forms