Octonions, triality, the exceptional Lie algebra F4, and polar actions on the Cayley hyperbolic plane

2018/02/22 by Andreas Kollross, Kollross, Andreas · 1 citation
Mathematics · Physics and Astronomy · #17A35 #53C35 #57S20 #Advanced Topics in Algebra #Differential Geometry (math.DG) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Nonlinear Waves and Solitons #Rings and Algebras (math.RA) #math.DG #math.RA #msc:17A35 #msc:53C35 #msc:57S20

paper · pdf · doi:10.48550/arxiv.1802.08075

v1: 24 pages; comments are welcome; v2: 25 pages, substantial revision, inconsistencies and typos corrected, new Lemma 3.2 added, notation simplified

openalex publication_date 2018/02/22 · arxiv created 2018/05/28 · arxiv updated 2018/05/30 · openalex created_date 2022/10/05 · openalex updated_date 2026/07/28

Abstract

Using octonions and the triality property of Spin(8), we find explicit formulae for the Lie brackets of the exceptional simple real Lie algebras \mathfrakf₄ and \mathfrakf^*₄, i.e. the Lie algebras of the isometry groups of the Cayley projective plane and the Cayley hyperbolic plane. As an application, we classify polar actions on the Cayley hyperbolic plane which leave a totally geodesic subspace invariant.

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