Lie supergroup actions on supermanifolds

Abstract

Lie supergroups are here understood as group objects in the category of supermanifolds (as in [ 2 2 , 5 5 , and 15 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 2 , 5 5 , and 12 12 ] and the natural actions on the Grassmannian supermanifolds studied in [ 6 6 - 9 9 and 13 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 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.