Realization of Lie superalgebras G(3) and F(4) as symmetries of supergeometries

Abstract

For every parabolic subgroup P of a Lie supergroup G the homogeneous superspace $G/P$ carries a G-invariant supergeometry. We address the problem whether $g=\mathrm{Lie}\left(G\right)$ is the maximal (local and global) symmetry of this supergeometry in the case of exceptional Lie superalgebras $G\left(3\right)$ and $F\left(4\right)$. Our approach is to consider the negatively graded Lie superalgebras for every choice of parabolic, and to compute the Tanaka-Weisfeiler prolongations, with reduction of the structure group when required (2 resp 3 special cases), thus realizing $G\left(3\right)$ and $F\left(4\right)$ as symmetries of supergeometries. This gives 19 inequivalent $G\left(3\right)$-supergeometries and 55 inequivalent $F\left(4\right)$-supergeometries, in majority of cases (17 resp 52 cases) those being encoded as vector superdistributions. We describe those supergeometries and realize supersymmetry explicitly.