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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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 [$ 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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Article copyright: © Copyright 1991 American Mathematical Society