2023/02/27 by Nurowski, Pawel
#Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.2302.13606
In Cartan's PhD thesis, there is a formula defining a certain rank 8 vector distribution in dimension 15, whose algebra of authomorphism is the split real form of the simple exceptional complex Lie algebra \mathfrakf4. Cartan's formula is written in the standard Cartesian coordinates in ℝ15. In the present paper we explain how to find analogous formula for the flat models of any bracket generating distribution \mathcal D whose symbol algebra \mathfrakn(\mathcal D) is constant and 2-step graded, \mathfrakn(\mathcal D)=\mathfrakn-2⊕\mathfrakn-1. The formula is given in terms of a solution to a certain system of linear algebraic equations determined by two representations (ρ,\mathfrakn-1) and (τ,\mathfrakn-2) of a Lie algebra \mathfrakn00 contained in the 0th order Tanaka prolongation \mathfrakn0 of \mathfrakn(\mathcal D). Numerous examples are provided, with particular emphasis on the distributions with symmetries being real forms of simple exceptional Lie algebras \mathfrakf4 and \mathfrake6.