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

SIGMA 11 (2015), 034, 12 pages      arXiv:1410.2339

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

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.

pdf (389 kb)   tex (19 kb)


  1. Demazure M., Grothendieck A., Schémas en groups. III. Structure des schémas en groupes reductifs, Séminaire de géométrie algébrique du Bois Marie, 1962/1964, available at
  2. Harder G., Bericht über neuere Resultate der Galoiskohomologie halbeinfacher Gruppen, Jber. Deutsch. Math.-Verein. 70 (1968), 182-216, available at
  3. Kneser M., Lectures on Galois cohomology of classical groups, Tata Institute of Fundamental Research Lectures on Mathematics, Vol. 47, Tata Institute of Fundamental Research, Bombay, 1969, available at publ/ln/tifr47.pdf.
  4. Knus M.A., Merkurjev A., Rost M., Tignol J.P., The book of involutions, American Mathematical Society Colloquium Publications, Vol. 44, Amer. Math. Soc., Providence, RI, 1998.
  5. Lam T.Y., Introduction to quadratic forms over fields, Graduate Studies in Mathematics, Vol. 67, Amer. Math. Soc., Providence, RI, 2005.
  6. Morris D., Real representations of semisimple Lie algebras have ${\mathbb Q}$-forms, in Algebraic Groups and Arithmetic, Tata Institute of Fundamental Research, Mumbai, 2004, 469-490.
  7. Platonov V., Rapinchuk A., Algebraic groups and number theory, Pure and Applied Mathematics, Vol. 139, Academic Press, Inc., Boston, MA, 1994.
  8. Prasad G., Rapinchuk A., On the existence of isotropic forms of semi-simple algebraic groups over number fields with prescribed local behavior, Adv. Math. 207 (2006), 646-660.
  9. Raghunathan M.S., Arithmetic lattices in semisimple groups, Proc. Indian Acad. Sci. Math. Sci. 91 (1982), 133-138.
  10. Tits J., Classification of algebraic semisimple groups, in Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), Amer. Math. Soc., Providence, R.I., 1966, 33-62.
  11. Tits J., Représentations linéaires irréductibles d'un groupe réductif sur un corps quelconque, J. Reine Angew. Math. 247 (1971), 196-220.
  12. Tits J., Strongly inner anisotropic forms of simple algebraic groups, J. Algebra 131 (1990), 648-677.

Previous article  Next article   Contents of Volume 11 (2015)