Lie supergroup actions on supermanifolds

Authors:
Charles P. Boyer and O. A. Sánchez-Valenzuela

Journal:
Trans. Amer. Math. Soc. **323** (1991), 151-175

MSC:
Primary 58A50; Secondary 17A70, 22E30, 22E60

MathSciNet review:
998351

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Abstract: Lie supergroups are here understood as group objects in the category of supermanifolds (as in [, , and ]). 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 [, , and ] and the natural actions on the Grassmannian supermanifolds studied in [- and ]. 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 []. 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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DOI:
https://doi.org/10.1090/S0002-9947-1991-0998351-4

Article copyright:
© Copyright 1991
American Mathematical Society