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Lie algebroids, holonomy and characteristic classes. (English) Zbl 1007.22007
From the author’s abstract: “We extend the notion of connection in order to study singular geometric structures, namely, we consider a notion of connection on a Lie algebroid which is a natural extension of the usual concept of a covariant connection. It allows us to define holonomy of the orbit foliation of a Lie algebroid and to prove a stability theorem. We also introduce secondary or exotic characteristic classes, thus providing invariants which generalize the modular class of a Lie algebroid”.

##### MSC:
 22A22 Topological groupoids (including differentiable and Lie groupoids) 53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) 17B99 Lie algebras and Lie superalgebras
##### Keywords:
Lie algebroid; connection; holonomy; characteristic
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##### References:
 [1] Bott, R., Lectures on characteristic classes and foliations, Lectures on algebraic and differential topology, 279, (1972), Springer-Verlag Berlin · Zbl 0241.57010 [2] Cannas da Silva, A.; Weinstein, A., Geometric models for noncommutative algebras, (1999), Amer. Math. Soc Providence · Zbl 1135.58300 [3] Chern, S.S.; Simons, J., Characteristic forms and geometric invariants, Ann. math., 99, 48-69, (1974) · Zbl 0283.53036 [4] Courant, T., Dirac manifolds, Trans. amer. math. soc., 319, 631-661, (1990) · Zbl 0850.70212 [5] M. Crainic, Differentiable and algebroid cohomology, Van Est isomorphisms, and characteristic classes, Comment. Math. Helv, to appear; preprint math.DG/0008064. [6] M. Crainic, Connections up to homotopy and characteristic classes, preprint math.DG/0010085. [7] M. Crainic, and, R. L. Fernandes, Integrability of Lie brackets, Ann. Math, to appear (also preprint math.DG/0105033). [8] Dazord, P., Feuilletages à singularités, Indag. math., 47, 21-39, (1985) · Zbl 0584.57016 [9] Evens, S.; Lu, J.-H.; Weinstein, A., Transverse measures, the modular class and a cohomology pairing for Lie algebroids, Quart. J. math. Oxford 2, 50, 417-436, (1999) · Zbl 0968.58014 [10] Fernandes, R.L., Connections in Poisson geometry I: holonomy and invariants, J. differential geom., 54, 303-366, (2000) · Zbl 1036.53060 [11] Ginzburg, V., Grothendieck groups of Poisson vector bundles, J. symplectic geom., 1, 121-169, (2001) · Zbl 1032.53072 [12] Ginzburg, V.; Golubev, A., Holonomy on Poisson manifolds and the modular class, Israel J. math., 122, 221-242, (2001) · Zbl 0991.53055 [13] Goldman, R., The holonomy ring on the leaves of foliated manifolds, J. differential geom., 11, 411-449, (1976) · Zbl 0356.57016 [14] Hermann, R., The differential geometry of foliations, II, J. math. mech., 11, 303-315, (1962) · Zbl 0152.20502 [15] Huebschmann, J., Lie-rinehart algebras, gerstenhaber algebras and Batalin-vilkovski algebras, Ann. inst. Fourier, 48, 425-440, (1998) · Zbl 0973.17027 [16] Huebschmann, J., Duality for Lie-rinehart algebras and the modular class, J. reine angew. math., 510, 103-159, (1999) · Zbl 1034.53083 [17] Itskov, V.; Karasev, M.; Vorobjev, Y., Infinitesimal Poisson cohomology, Amer. math. soc. transl. (2), 187, 327-360, (1998) · Zbl 0922.58028 [18] Kamber, F.; Tondeur, P., Foliated bundles and characteristic classes, (1975), Springer-Verlag Berlin/New York · Zbl 0308.57011 [19] Kerbrat, Y.; Souici-Benhammadi, Z., Variétés de Jacobi et groupoı̈des de contact, C. R. acad. sci. ser. I, 317, 81-86, (1993) · Zbl 0804.58015 [20] Kobayashi, S.; Nomizu, K., Foundations of differential geometry, (1969), Interscience Publ New York · Zbl 0175.48504 [21] Kosmann-Schwarzbach, Y., Modular vector fields and Batalin-Vilkovisky algebras, Poisson geometry, Banach center publications, 51, (2000), Institute of MathematicsPolish Academy of Sciences Warszawa · Zbl 1018.17020 [22] J. Kubarski, Characteristic classes of regular Lie algebroids—a sketch, inProceedings of the Winter School “Geometry and Physics” (Srni, 1991), Rend. Circ. Mat. Palermo (2)301993, 71-94. · Zbl 0804.57016 [23] Kubarski, J., Bott’s vanishing theorem for regular Lie algebroids, Trans. amer. math. soc., 348, 2151-2167, (1996) · Zbl 0858.22009 [24] Mackenzie, K., Lie groupoids and Lie algebroids in differential geometry, (1987), Cambridge Univ. Press Cambridge · Zbl 0683.53029 [25] Mackenzie, K., Lie algebroids and Lie pseudoalgebras, Bull. London math. soc., 27, 97-147, (1995) · Zbl 0829.22001 [26] Moore, C.; Schochet, C., Global analysis on foliated spaces, (1988), Mathematical Sciences Research Institute Publications · Zbl 0648.58034 [27] Sussmann, H., Orbits of families of vector fields and integrability of distributions, Trans. amer. math. soc., 180, 171-188, (1973) · Zbl 0274.58002 [28] Vaisman, I., On the geometric quantization of Poisson manifolds, J. math. phys., 32, 3339-3345, (1991) · Zbl 0749.58023 [29] Vaisman, I., Lectures on the geometry of Poisson manifolds, (1994), Birkhäuser Berlin · Zbl 0852.58042 [30] Weinstein, A., The local structure of Poisson manifolds, J. differential geom., 18, 523-557, (1983) · Zbl 0524.58011 [31] Weinstein, A., Lagrangian mechanics and groupoids, Mechanics day (Waterloo, ON, 1992), 7, (1996), Fields Institute of Communication, p. 207-231 · Zbl 0844.22007 [32] Weinstein, A., The modular automorphism group of a Poisson manifold, J. geom. phys., 23, 379-394, (1997) · Zbl 0902.58013 [33] Weinstein, A., Linearization problems for Lie algebroids and Lie groupoids, Lett. math. phys., 52, 93-102, (2000) · Zbl 0961.22004 [34] Xu, P., Gerstenhaber algebras and BV-algebras in Poisson geometry, Comm. math. phys., 200, 545-560, (1999) · Zbl 0941.17016
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