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A Cohomological Proof that Real Representations of Semisimple Lie Algebras Have $\mathbb{Q}$-Forms
Symmetry, integrability and geometry: methods and applications, Tome 11 (2015) Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: semisimple Lie algebra; finite-dimensional representation; global field; Galois cohomology; linear algebraic group; Tits algebra. 
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Dave Witte Morris. A Cohomological Proof that Real Representations of Semisimple Lie Algebras Have $\mathbb{Q}$-Forms. Symmetry, integrability and geometry: methods and applications, Tome 11 (2015). http://geodesic.mathdoc.fr/item/SIGMA_2015_11_a33/

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