2022/07/28 by de Traubenberg, Michel Rausch, Slupinski, M J
#Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2207.13946
We continue our study and classification of structures on the Fano plane \cal F and its dual \cal F^∗ involved in the construction of octonions and the Lie algebra \mathfrak g2 (\mathbb F) over a field \mathbb F. These are a "composition factor": \cal F× \cal F →\-1, 1\, inducing an octonion multiplication, and a function δ^∗ : Aut(\cal F) × \cal F^∗ → \-1, 1\ such that g ∈ Aut(\cal F) can be lifted to an automorphism of the octonions iff δ^∗(g, ⋅) is the Radon transform of a function on \cal F. We lift the action of Aut(\cal F) on \cal F to the action of a non-trivial eight-fold covering Aut(\cal F) on a twofold covering \cal F of \cal F contained in the octonions. This extends tautologically to an action on the octonions by automorphism. Finally, we associate to incident point-line pairs a generating set of \mathfrak g2 (\mathbb F) and express brackets in terms of the incidence geometry of \cal F and ε.