On the compact real forms of the Lie algebras of type E₆ and F₄

2012/08/20 by Robert A. Wilson, Wilson, Robert A.
Mathematics · #17B25 #17B45 #20G20 #20G40 #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Rings and Algebras (math.RA) #math.GR #math.RA #msc:17B25 #msc:17B45 #msc:20G20 #msc:20G40

paper · pdf · doi:10.48550/arxiv.1208.3967

arxiv created 2012/08/20 · openalex publication_date 2012/08/20 · arxiv updated 2012/08/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We give a construction of the compact real form of the Lie algebra of type E₆, using the finite irreducible subgroup of shape 3³⁺³:SL₃(3), which is isomorphic to a maximal subgroup of the orthogonal group Ω₇(3). In particular we show that the algebra is uniquely determined by this subgroup. Conversely, we prove from first principles that the algebra satisfies the Jacobi identity, and thus give an elementary proof of existence of a Lie algebra of type E₆. The compact real form of F₄ is exhibited as a subalgebra.

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