Geometric quantization of Hamiltonian actions of Lie algebroids and Lie groupoids. (English) Zbl 1140.22002
This paper introduces a method to construct Hermitian representations of Lie algebroids by a geometric quantization procedure. Lie groupoids play an important role in the study of manifolds with boundaries or corners and in Poisson geometry. In this work, the author generalizes the construction of representations to the case of Lie algebroids and Lie groupoids. He first recalls the relevant notions, namely the notion of a Lie groupoid or a Lie algebroid action on a map. Then he constructs a prequantization line bundle. The main theorem follows. Finally, it is showed how the well-known orbit method, developped by Kirillov, and the nice results for reductive Lie groups can be extended for Lie groupoids.
Reviewer: Angela Gammella-Mathieu (Metz)
MSC:
22A22 | Topological groupoids (including differentiable and Lie groupoids) |
53D20 | Momentum maps; symplectic reduction |
58H05 | Pseudogroups and differentiable groupoids |
53D50 | Geometric quantization |
57R30 | Foliations in differential topology; geometric theory |
Keywords:
geometric quantization; Lie algebroids; Lie groupoids; representation theory; hamiltonian action; orbit methodReferences:
[1] | Bott R., Graduate Texts in Mathematics 82 (1982) · doi:10.1007/978-1-4757-3951-0 |
[2] | Connes A., Noncommutative Geometry (1994) |
[3] | Corwin L. J., Cambridge Studies in Advanced Mathematics 18, in: Representations of Nilpotent Lie Groups and Their Applications. Part I. Basic Theory and Examples (1990) |
[4] | DOI: 10.1007/s00014-001-0766-9 · Zbl 1041.58007 · doi:10.1007/s00014-001-0766-9 |
[5] | DOI: 10.4007/annals.2003.157.575 · Zbl 1037.22003 · doi:10.4007/annals.2003.157.575 |
[6] | DOI: 10.1093/qjmath/50.200.417 · Zbl 0968.58014 · doi:10.1093/qjmath/50.200.417 |
[7] | DOI: 10.1007/BF01398934 · Zbl 0503.58018 · doi:10.1007/BF01398934 |
[8] | DOI: 10.1090/surv/098 · doi:10.1090/surv/098 |
[9] | Kirillov A. A., Lectures on the Orbit Method, Graduate Studies in Mathematics 64 (2004) · Zbl 1229.22003 |
[10] | DOI: 10.1017/CBO9780511614156.017 · doi:10.1017/CBO9780511614156.017 |
[11] | DOI: 10.1007/978-1-4612-1680-3 · doi:10.1007/978-1-4612-1680-3 |
[12] | DOI: 10.1007/978-3-0348-8364-1_7 · doi:10.1007/978-3-0348-8364-1_7 |
[13] | DOI: 10.1142/9789812701244_0002 · doi:10.1142/9789812701244_0002 |
[14] | DOI: 10.1017/CBO9781107325883 · doi:10.1017/CBO9781107325883 |
[15] | DOI: 10.1016/0034-4877(74)90021-4 · Zbl 0327.58005 · doi:10.1016/0034-4877(74)90021-4 |
[16] | DOI: 10.1006/aima.1997.1701 · Zbl 0929.53045 · doi:10.1006/aima.1997.1701 |
[17] | DOI: 10.1016/S0040-9383(98)00012-3 · Zbl 0928.37013 · doi:10.1016/S0040-9383(98)00012-3 |
[18] | DOI: 10.2977/prims/1195175328 · Zbl 0659.58016 · doi:10.2977/prims/1195175328 |
[19] | DOI: 10.1353/ajm.2002.0019 · Zbl 1013.58010 · doi:10.1353/ajm.2002.0019 |
[20] | DOI: 10.1017/CBO9780511615450 · doi:10.1017/CBO9780511615450 |
[21] | DOI: 10.1006/jfan.2001.3825 · Zbl 1001.53062 · doi:10.1006/jfan.2001.3825 |
[22] | DOI: 10.1090/conm/282/04682 · doi:10.1090/conm/282/04682 |
[23] | DOI: 10.1007/s002220050223 · Zbl 0944.53047 · doi:10.1007/s002220050223 |
[24] | DOI: 10.1023/A:1007744304422 · Zbl 0939.19001 · doi:10.1023/A:1007744304422 |
[25] | Tu J. L., Ann. Sci. Ecole Norm. Sup. (4) 37 pp 841– |
[26] | DOI: 10.1016/B978-012625440-2/50006-9 · doi:10.1016/B978-012625440-2/50006-9 |
[27] | DOI: 10.1090/S0273-0979-1987-15473-5 · Zbl 0618.58020 · doi:10.1090/S0273-0979-1987-15473-5 |
[28] | Weinstein A., Notices Amer. Math. Soc. 43 pp 744– |
[29] | Weinstein A., J. Reine Angew. Math. 417 pp 159– |
[30] | Woodhouse N. M. J., Oxford Mathematical Monographs, in: Geometric Quantization (1992) |
[31] | Zhang Weiping, Int. Math. Res. Not. 19 pp 1043– |
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.