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Journal of Lie Theory 26 (2016), No. 2, 297--358
Copyright Heldermann Verlag 2016

Integrating Infinitesimal (Super) Actions

Gijs M. Tuynman
Laboratoire Paul Painlevé, UFR de Mathématiques, Université de Lille I, F-59655 Villeneuve d'Ascq, France

We generalize some results of Richard Palais to the case of Lie supergroups and Lie superalgebras. More precisely, let G be a Lie supergroup, g its Lie superalgebra and let ρ be an infinitesimal action (a representation) of g on a supermanifold M. We will show that there always exists a local (smooth left) action of G on M such that ρ is the map that associates the fundamental vector field on M to an algebra element (we will say that the action integrates ρ). We also show that if ρ is univalent, then there exists a unique maximal local action of G on M integrating ρ. And finally we show that if G is simply connected and all (smooth, even) vector fields ρ(X) are complete then there exists a global (smooth left) action of G on M integrating ρ. Omitting all references to the super setting will turn our proofs into variations of those of Palais.

Keywords: Supermanifolds, Lie superalgebras, Lie supergroups, infinitesimal local group actions.

MSC: 58A50, 57S20, 58C50

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