×

On the integrability of Lie subalgebroids. (English) Zbl 1131.58015

It is well known that if \(G\) is a Lie group with the Lie algebra \(\mathfrak{g}\), then every Lie subalgebra \(\mathfrak{h}\) of \(\mathfrak{g}\) can be integrated by a Lie subgroup of \(G\). The analogous fact holds for transitive Lie algebroids and Lie groupoids [see J. Kubarski, Colloq. Math. 54, 39–48 (1987; Zbl 0638.22001)]. However, this is not true in the category of nontransitive Lie algebroids. In the previous paper of the authors [Am. J. Math. 124, No. 3, 567–593 (2002; Zbl 1013.58010)] it is proved that any Lie subalgebroid \(\mathfrak{h}\) of an integrable Lie algebroid \(\mathfrak{g}\) is itself integrable, while the inclusion \(\mathfrak{h}\hookrightarrow\mathfrak{g}\) integrates to an immersion \(H\rightarrow G\) of the corresponding Lie groupoids, but there may not exist such an injective immersion.
The aim of the paper is to give conditions under which the integration of \(\mathfrak{h}\) to a Lie subgroupoid \(H\) of \(G\mathfrak{\;}\)is possible. The problem of integrability by closed subgroupoids is also solved. One of the main theorems says that a Lie subalgebroid \(\mathfrak{h}\subset\mathfrak{g}\) (\(\mathfrak{g}\) is assumed to be integrated to a Lie groupoid \(G\)) can be integrated to an injective immersion \(H\rightarrow G\) if and only if the right invariant foliation of \(G\) determined by \(\mathfrak{h}\) has a trivial holonomy.

MSC:

58H05 Pseudogroups and differentiable groupoids
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Almeida, R.; Molino, P., Suites d’Atiyah et feuilletages transversalement complets, C. R. Acad. Sci. Paris Sér. I Math., 300, 1, 13-15 (1985) · Zbl 0582.57015
[2] H. Bursztyn, A. Weinstein, Poisson Geometry, Deformation Quantisation and Group Representations, LMS Lecture Notes Series 323, pp. 1-78, Cambridge University Press, Cambridge, 2005.; H. Bursztyn, A. Weinstein, Poisson Geometry, Deformation Quantisation and Group Representations, LMS Lecture Notes Series 323, pp. 1-78, Cambridge University Press, Cambridge, 2005. · Zbl 1065.53001
[3] Cannas da Silva, A.; Weinstein, A., Geometric Models for Noncommutative Algebras, (Berkeley Mathematics Lecture Notes, vol. 10 (1999), American Mathematical Society: American Mathematical Society Providence, RI) · Zbl 1135.58300
[4] P. Cartier, Galois theory for differential equations—a geometric approach, Lecture at Utrecht University, 2004.; P. Cartier, Galois theory for differential equations—a geometric approach, Lecture at Utrecht University, 2004.
[5] Cattaneo, A. S.; Felder, G., A path integral approach to the Kontsevich quantization formula, Comm. Math. Phys., 212, 3, 591-611 (2000) · Zbl 1038.53088
[6] Conlon, L., Transversally parallelizable foliations of codimension two, Trans. Amer. Math. Soc., 194, 79-102 (1974) · Zbl 0288.57011
[7] Connes, A., Noncommutative Geometry (1994), Academic Press: Academic Press San Diego, CA · Zbl 0681.55004
[8] Crainic, M.; Fernandes, R. L., Integrability of Lie brackets, Ann. Math. (2), 157, 2, 575-620 (2003) · Zbl 1037.22003
[9] Crainic, M.; Moerdijk, I., Foliation groupoids and their cyclic homology, Adv. Math., 157, 2, 177-197 (2001) · Zbl 0989.22010
[10] Fedida, E., Sur les feuilletages de Lie, C. R. Acad. Sci. Paris Sér. A-B, 272, A999-A1001 (1971) · Zbl 0218.57014
[11] Higgins, P. J.; Mackenzie, K., Algebraic constructions in the category of Lie algebroids, J. Algebra, 129, 1, 194-230 (1990) · Zbl 0696.22007
[12] S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, vol. I, Wiley Classics Library, Wiley, New York, 1996. Reprint of the 1963 original, A Wiley-Interscience Publication.; S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, vol. I, Wiley Classics Library, Wiley, New York, 1996. Reprint of the 1963 original, A Wiley-Interscience Publication. · Zbl 0119.37502
[13] A. Kumpera, D. Spencer, Lie Equations, vol. I: General Theory. Princeton University Press, Princeton, NJ, 1972, Ann. Math. Stud. No. 73.; A. Kumpera, D. Spencer, Lie Equations, vol. I: General Theory. Princeton University Press, Princeton, NJ, 1972, Ann. Math. Stud. No. 73. · Zbl 0258.58015
[14] Landsman, N. P., Mathematical Topics between Classical and Quantum Mechanics, (Springer Monographs in Mathematics (1998), Springer: Springer New York) · Zbl 0923.00008
[15] Mackenzie, K., Lie Groupoids and Lie Algebroids in Differential Geometry, (London Mathematical Society Lecture Note Series, vol. 124 (1987), Cambridge University Press: Cambridge University Press Cambridge) · Zbl 0683.53029
[16] Mackenzie, K. C.H.; Xu, P., Integration of Lie bialgebroids, Topology, 39, 3, 445-467 (2000) · Zbl 0961.58009
[17] Moerdijk, I.; Mrčun, J., On integrability of infinitesimal actions, Amer. J. Math., 124, 3, 567-593 (2002) · Zbl 1013.58010
[18] Moerdijk, I.; Mrčun, J., Introduction to Foliations and Lie Groupoids, (Cambridge Studies in Advanced Mathematics, vol. 91 (2003), Cambridge University Press: Cambridge University Press Cambridge) · Zbl 1029.58012
[19] I. Moerdijk, J. Mrčun, On the developability of subalgebroids, preprint, arXiv:math.DG/0406561, 2004.; I. Moerdijk, J. Mrčun, On the developability of subalgebroids, preprint, arXiv:math.DG/0406561, 2004.
[20] Molino, P., Étude des feuilletages transversalement complets et applications, Ann. Sci. École Norm. Sup. (4), 10, 3, 289-307 (1977) · Zbl 0368.57007
[21] P. Molino, Riemannian Foliations, Progress in Mathematics, vol. 73, Birkhäuser Boston Inc., Boston, MA, 1988. (Translated from the French by Grant Cairns, With appendices by Cairns, Y. Carrière, É. Ghys, E. Salem and V. Sergiescu.); P. Molino, Riemannian Foliations, Progress in Mathematics, vol. 73, Birkhäuser Boston Inc., Boston, MA, 1988. (Translated from the French by Grant Cairns, With appendices by Cairns, Y. Carrière, É. Ghys, E. Salem and V. Sergiescu.) · Zbl 0633.53001
[22] Pradines, J., Théorie de Lie pour les groupoı¨des différentiables. Calcul différenetiel dans la catégorie des groupoı¨des infinitésimaux, C. R. Acad. Sci. Paris Sér. A-B, 264, A245-A248 (1967) · Zbl 0154.21704
[23] J.-P. Serre, Lie Algebras and Lie Groups, vol. 1500, second ed., Lecture Notes in Mathematics, Springer, Berlin, 1992 (1964 lectures given at Harvard University.); J.-P. Serre, Lie Algebras and Lie Groups, vol. 1500, second ed., Lecture Notes in Mathematics, Springer, Berlin, 1992 (1964 lectures given at Harvard University.)
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.