Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 034, 12 pages      arXiv:1410.2339      https://doi.org/10.3842/SIGMA.2015.034

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

Dave Witte Morris
Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, Alberta, T1K 3M4, Canada

Received October 17, 2014, in final form April 14, 2015; Published online April 27, 2015

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.

Key words: semisimple Lie algebra; finite-dimensional representation; global field; Galois cohomology; linear algebraic group; Tits algebra.

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