×

On Lie algebroid actions and morphisms. (English) Zbl 0873.58072

Author’s abstract: “Let \(AG\) be the Lie algebroid of a Lie groupoid \(G\). We prove that every infinitesimal action of \(AG\) on a manifold \(M\) by fundamental vector fields completes, can be lifted to a unique action of the Lie groupoid \(G\) on \(M\), if the \(\alpha\)-fibres of \(G\) are connected and simply connected. We apply this result to find the integration of Lie algebroid morphisms, when the common base of these algebroids is connected”.

MSC:

58H05 Pseudogroups and differentiable groupoids
22A22 Topological groupoids (including differentiable and Lie groupoids)
20L05 Groupoids (i.e. small categories in which all morphisms are isomorphisms)
22E60 Lie algebras of Lie groups
PDFBibTeX XMLCite
Full Text: Numdam EuDML

References:

[1] C. Albert and P. Dazord , Théorie genérale des groupoïdes de Lie , Publication du département de mathématiques de l’université de Lyon 1 , 53 - 105 , 1989 .
[2] R. Almeida and A. Kumpera , Structure produit dans la catégorie des algébroïdes de Lie , An. Acad. brasil. Ciênc. , 53 ( 2 ), 247 - 250 , 1981 . MR 637370 | Zbl 0484.58033 · Zbl 0484.58033
[3] R. Brown and O. Mucuk , The monodromy groupoid of a Lie groupoid , Cah. Top. Géom. Diff. Cat. , Vol XXXVI - 4 , 345 - 369 , 1995 . Numdam | MR 1367591 | Zbl 0844.22006 · Zbl 0844.22006
[4] P. Dazord , Groupoïdes symplectiques et troisiéme théoréme de Lie ”non linéaire ” . Lecture Notes in Mathematics , vol. 1416 . Berlin , Heidelberg , New York : Springer , pp. 39 - 44 , 1990 . MR 1047476 | Zbl 0702.58023 · Zbl 0702.58023
[5] E. Pourreza , thése 3em Cycle , Toulouse , 1972 .
[6] P.J. Higgins and K. Mackenzie , Algebraic constructions in the category of Lie algebroids, Journal of Algebra 129 , 194 - 230 , 1990 . MR 1037400 | Zbl 0696.22007 · Zbl 0696.22007 · doi:10.1016/0021-8693(90)90246-K
[7] A. Kumpera and D.C. Spencer , Lie equations, volume 1: General theory , Princeton University Press , 1972 . Zbl 0258.58015 · Zbl 0258.58015
[8] K. Mackenzie , Lie groupoids and Lie algebroids in differential geometry , London Mathematical Society Lecture Note Series , Vol 124 , Cambridge Univ Press , Cambridge . MR 896907 | Zbl 0683.53029 · Zbl 0683.53029
[9] K. Mackenzie and P. Xu , Integration of Lie bialgebroids , preprint. arXiv | MR 1746902 | Zbl 0961.58009 · Zbl 0961.58009 · doi:10.1016/S0040-9383(98)00069-X
[10] T. Mokri , PhD thesis , University of Sheffield , 1995 .
[11] P. Molino , Riemannian foliations , Birkhauser , Boston , 1988 . MR 932463 | Zbl 0633.53001 · Zbl 0633.53001
[12] R. Palais , Global formulation of the Lie theory of transformation groups , Memoirs Amer. Math. Soc. , 23 , 1 - 123 , 1957 . MR 121424 | Zbl 0178.26502 · Zbl 0178.26502
[13] J. Pradines , Théorie de Lie pour les groupoïdes différentiables. Relations entre les propriétés locales et globales , C. R. Acad. Sci. Paris Sér. A 263 , 907 - 910 , 1966 . MR 214103 | Zbl 0147.41102 · Zbl 0147.41102
[14] J. Pradines , Théorie de Lie pour les groupoïdes différentiables. Calcul différentiel dans la catégorie des groupoïdes infinitésimaux , C. R. Acad. Sci. Paris Sér. A 264 , 245 - 248 , 1967 . MR 216409 | Zbl 0154.21704 · Zbl 0154.21704
[15] J. Pradines , Géométrie différentielle au dessus d’un groupoïde , C. R. Acad. Sci. Paris Sér. A 266 , 1194 - 1196 , 1967 . MR 231306 | Zbl 0172.03601 · Zbl 0172.03601
[16] P. Stefan , Accessible sets, orbits, and foliations with singularities , Bulletin Amer. Math. Soc. , 80 , ( 6 ), 1142 - 1145 , 1974 . Article | MR 353362 | Zbl 0293.57015 · Zbl 0293.57015 · doi:10.1090/S0002-9904-1974-13648-7
[17] H.J. Sussmann , Orbits of families of vector fields and integrability of distributions , Transactions of the Amer. math. Soc. , vol 180 , 171 - 188 , 1973 . MR 321133 | Zbl 0274.58002 · Zbl 0274.58002 · doi:10.2307/1996660
[18] V S. Varadarajan, Lie groups, Lie algebras, and their representations , Springer-Verlag , 1984 . MR 746308 | Zbl 0955.22500 · Zbl 0955.22500
[19] P. Xu , Morita equivalence of symplectic groupoids , Commun. Math. Phys. 142 , 493 - 509 , 1991 . Article | Zbl 0746.58034 · Zbl 0746.58034 · doi:10.1007/BF02099098
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.