Root space decomposition of \mathfrakg₂ from octonions

2017/08/08 by Basak, Tathagata
#16W25 (Primary) 16W10 #17B25 (Secondary) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1708.02367

Abstract

We describe a simple way to write down explicit derivations of octonions that form a Chevalley basis of \mathfrakg₂. This uses the description of octonions as a twisted group algebra of the finite field \mathbbF₈. Generators of Gal(\mathbbF₈/\mathbbF₂) act on the roots as 120-degree rotations and complex conjugation acts as negation.

Related

• The octonions as a twisted group algebra
• Commutation relations of \mathfrak g_2 and the incidence geometry of the Fano plane
• Lattice Filtrations for G₂ of a p-adic Field
• Polynomial Equations over Octonion Algebras
• Quasialgebra structure of the octonions