Integrating Infinitesimal (Super) Actions
Journal of Lie theory, Tome 26 (2016) no. 2, pp. 297-358
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.
Classification :
58A50, 57S20, 58C50
Mots-clés : Supermanifolds, Lie superalgebras, Lie supergroups, infinitesimal local group actions
Mots-clés : Supermanifolds, Lie superalgebras, Lie supergroups, infinitesimal local group actions
@article{JLT_2016_26_2_JLT_2016_26_2_a0,
author = {G. M. Tuynman},
title = {Integrating {Infinitesimal} {(Super)} {Actions}},
journal = {Journal of Lie theory},
pages = {297--358},
year = {2016},
volume = {26},
number = {2},
url = {http://geodesic.mathdoc.fr/item/JLT_2016_26_2_JLT_2016_26_2_a0/}
}
G. M. Tuynman. Integrating Infinitesimal (Super) Actions. Journal of Lie theory, Tome 26 (2016) no. 2, pp. 297-358. http://geodesic.mathdoc.fr/item/JLT_2016_26_2_JLT_2016_26_2_a0/