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The exceptional simple Lie group $F_{4(-20)}$, after J. Tits

Alain J. Valette

Vol. 20 (2023), No. 2-3, 599–610

This is a semisurvey paper, where we start by advertising Tits’ synthetic construction (1953) of the hyperbolic plane H2(Cay ) over the Cayley numbers Cay and of its automorphism group which is the exceptional simple Lie group G = F4(20). Let G = KAN be the Iwasawa decomposition. Our contributions are:

  • Writing down explicitly the action of N on H2(Cay ) in Tits’ model, facing the lack of associativity of Cay .

  • If MAN denotes the minimal parabolic subgroup of G, characterizing M geometrically.

exceptional simple Lie groups, octonions, hyperbolic plane
Mathematical Subject Classification
Primary: 22E15
Secondary: 17A35, 51A10, 51A45
Received: 17 November 2022
Revised: 27 March 2023
Accepted: 19 April 2023
Published: 13 September 2023
Alain J. Valette
Institut de mathématiques, Faculté des Sciences
Université de Neuchâtel