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

Boris KruglikovAndreu Llabres
Dec 2022
摘要
For every parabolic subgroup $P$ of a Lie supergroup $G$ the homogeneoussuperspace $G/P$ carries a $G$-invariant supergeometry. We address the quesitonwhether $\mathfrak{g}=\operatorname{Lie}(G)$ is the maximal symmetry of thissupergeometry in the case of exceptional Lie superalgebras $G(3)$ and $F(4)$.Our approach is to consider the negatively graded Lie superalgebras for everychoice of parabolic, and to compute the Tanaka-Weisfeiler prolongations, withreduction of the structure group when required (2 resp 3 cases), thus realizing$G(3)$ and $F(4)$ as symmetries of supergeometries. This gives 19 inequivalent$G(3)$-supergeometries and 55 inequivalent $F(4)$-supergeometries, in majorityof cases (17 resp 52 cases) those being encoded as vector superdistributions.We describe those supergeometries and realize supersymmetry explicitly in somecases.
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