Lie supergroup actions on supermanifolds

Boyer, Charles P.; Sánchez-Valenzuela, O. A.

doi:10.1090/S0002-9947-1991-0998351-4

Lie supergroup actions on supermanifolds
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by Charles P. Boyer and O. A. Sánchez-Valenzuela PDF

Trans. Amer. Math. Soc. 323 (1991), 151-175 Request permission

Abstract:

Lie supergroups are here understood as group objects in the category of supermanifolds (as in [$2$, $5$, and $15$]). Actions of Lie supergroups in supermanifolds are defined by means of diagrams of supermanifold morphisms. Examples of such actions are given. Among them emerge the linear actions discussed in [$2$, $5$, and $12$] and the natural actions on the Grassmannian supermanifolds studied in [$6$-$9$ and $13$]. The nature of the isotropy subsupergroup associated to an action is fully elucidated; it is exhibited as an embedded subsupergroup in the spirit of the theory of smooth manifolds and Lie groups and with no need for the Lie-Hopf algebraic approach of Kostant in [$3$]. The notion of orbit is also discussed. Explicit calculations of isotropy subsupergroups are included. Also, an alternative proof of the fact that the structural sheaf of a Lie supergroup is isomorphic to the sheaf of sections of a trivial exterior algebra bundle is given, based on the triviality of its supertangent bundle.

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