Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 4 (2008), 005, 30 pages      arXiv:0710.3098
Contribution to the Proceedings of the 2007 Midwest Geometry Conference in honor of Thomas P. Branson

Poisson Manifolds, Lie Algebroids, Modular Classes: a Survey

Yvette Kosmann-Schwarzbach
Centre de Mathématiques Laurent Schwartz, École Polytechnique, 91128 Palaiseau, France

Received August 31, 2007, in final form January 02, 2008; Published online January 16, 2008

After a brief summary of the main properties of Poisson manifolds and Lie algebroids in general, we survey recent work on the modular classes of Poisson and twisted Poisson manifolds, of Lie algebroids with a Poisson or twisted Poisson structure, and of Poisson-Nijenhuis manifolds. A review of the spinor approach to the modular class concludes the paper.

Key words: Poisson geometry; Poisson cohomology; modular classes; twisted Poisson structures; Lie algebroids; Gerstenhaber algebras; Lie algebroid cohomology; triangular r-matrices; quasi-Frobenius algebras; pure spinors.

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  1. Abouqateb A., Boucetta M., The modular class of a regular Poisson manifold and the Reeb class of its symplectic foliation, C. R. Math. Acad. Sci. Paris 337 (2003), 61-66, math.DG/0211405.
  2. Abraham R., Marsden J.E., Foundations of mechanics, W.A. Benjamin, Inc., New York - Amsterdam, 1967.
  3. Abraham R., Marsden J.E., Foundations of mechanics, second edition, revised and enlarged, with the assistance of Tudor Ratiu and Richard Cushman, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1978.
  4. Agrotis M., Damianou P., The modular hierarchy of the Toda lattice, Differential Geom. Appl., to appear, math.DG/0701057.
  5. Alekseev A., Bursztyn H., Meinrenken E., Pure spinors on Lie groups, arXiv:0709.1452.
  6. Alekseev A., Kosmann-Schwarzbach Y., Meinrenken E., Quasi-Poisson manifolds, Canad. J. Math. 54 (2002), 3-29, math.DG/0006168.
  7. Alekseev A., Xu P., Derived brackets and Courant algebroids, unfinished manuscript, 2000.
  8. Beltrán J.V., Monterde J., Poisson-Nijenhuis structures and the Vinogradov bracket, Ann. Global Anal. Geom. 12 (1994), 65-78.
  9. Bhaskara K.H., Viswanath K., Calculus on Poisson manifolds, Bull. London Math. Soc. 20 (1988), 68-72.
  10. Bojowald M., Kotov A., Strobl T., Lie algebroid morphisms, Poisson sigma models, and off-shell closed gauge symmetries, J. Geom. Phys. 54 (2005), 400-426, math.DG/0406445.
  11. Bonechi F., Zabzine M., Poisson sigma model on the sphere, arXiv:0706.3164.
  12. Brylinski J.-L., A differential complex for Poisson manifolds, J. Differential Geom. 28 (1988), 93-114.
  13. Brylinski J.-L., Zuckerman G., The outer derivation of a complex Poisson manifold, J. Reine Angew. Math. 506 (1999), 181-189, math.DG/9802014.
  14. Bursztyn H., Radko O., Gauge equivalence of Dirac structures and symplectic groupoids, Ann. Inst. Fourier (Grenoble) 53 (2003), 309-337, math.SG/0202099.
  15. Bursztyn H., Weinstein A., Picard groups in Poisson geometry, Mosc. Math. J. 4 (2004), 39-66, math.SG/0304048.
  16. Cannas da Silva A., Weinstein A., Geometric models for noncommutative algebras, Berkeley Mathematics Lecture Notes, Vol. 10, Amer. Math. Soc., Providence, RI, 1999.
  17. Caseiro R., Modular classes of Poisson-Nijenhuis Lie algebroids, Lett. Math. Phys. 80 (2007), 223-238, math.DG/0701476.
  18. Caseiro R., Nunes da Costa J. M., Jacobi-Nijenhuis algebroids and their modular classes, arXiv:0706.1475.
  19. Chari V., Pressley A., A guide to quantum groups, Cambridge University Press, Cambridge, 1995.
  20. Chen Z., Liu Z.-J., On (co-)morphisms of Lie pseudoalgebras and groupoids, J. Algebra 316 (2007), 1-31, arXiv:0710.2149.
  21. Chevalley C., The algebraic theory of spinors, Columbia University Press, New York, 1954; The algebraic theory of spinors and Clifford algebras, Collected Works, Vol. 2, Springer-Verlag, Berlin, 1997.
  22. Cornalba L., Schiappa R., Nonassociative star product deformations for D-brane world-volumes in curved backgrounds, Comm. Math. Phys. 225 (2002), 33-66, hep-th/0101219.
  23. Coste A., Dazord P., Weinstein A., Groupoïdes symplectiques, Publications du Département de Mathématiques, Université Claude Bernard-Lyon I 2A (1987), 1-62.
  24. Crainic M., Differentiable and algebroid cohomology, van Est isomorphisms, and characteristic classes, Comment. Math. Helv. 78 (2003), 681-721, math.DG/0008064.
  25. Damianou P.A., Fernandes R.L., Integrable hierarchies and the modular class, math.DG/0607784.
  26. Dolgushev V., The Van den Bergh duality and the modular symmetry of a Poisson variety, math.QA/0612288.
  27. Dorfman I.Ya., Dirac structures and integrability of nonlinear evolution equations, John Wiley and Sons, Chichester, 1993.
  28. Douady A., Lazard M., Espaces fibrés en algèbres de Lie et en groupes, Invent. Math. 1 (1966), 133-151.
  29. Drinfeld V.G., Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of classical Yang-Baxter equations, Dokl. Akad. Nauk SSSR 268 (1983), no. 2, 285-287 (English transl.: Sov. Math. Dokl. 27 (1983), no. 2, 68-71).
  30. Drinfeld V.G., Quasi-Hopf algebras, Algebra i Analiz 1 (1989), 114-148 (English transl.: Leningrad Math. J. 1 (1990), 1419-1457).
  31. Dufour J.-P., Haraki A., Rotationnels et structures de Poisson quadratiques, C. R. Acad. Sci. Paris Sér. I Math. 312 (1991), 137-140.
  32. Ehresmann C., Catégories topologiques et catégories différentiables, in Colloque Géom. Diff. Globale (1958, Bruxelles), Centre Belge Rech. Math., Louvain, 1959, 137-150.
  33. Evens S., Lu J.-H., Poisson harmonic forms, Kostant harmonic forms, and the S\sp1-equivariant cohomology of K/T, Adv. Math. 142 (1999), 171-220, dg-ga/9711019.
  34. Evens S., Lu J.-H., Weinstein A., Transverse measures, the modular class and a cohomology pairing for Lie algebroids, Quart. J. Math. Ser. 2 50 (1999), 417-436, dg-ga/9610008.
  35. Fernandes R., Connections in Poisson geometry. I. Holonomy and invariants, J. Differential Geom. 54 (2000), 303-365, math.DG/0001129.
  36. Fernandes R., Lie algebroids, holonomy and characteristic classes, Adv. Math. 170 (2002), 119-179, math.DG/0007132.
  37. Fernandes R., Invariants of Lie algebroids, Differential Geom. Appl. 19 (2003), 223-243, math.DG/0202254.
  38. Flato M., Lichnerowicz A., Sternheimer D., Algèbres de Lie attachées a une variété canonique, J. Math. Pures Appl. (9) 54 (1975), 445-480.
  39. Fuchssteiner B., Lie algebra structure of degenerate Hamiltonian and bi-Hamiltonian systems, Progr. Theoret. Phys. 68 (1982), 1082-1104.
  40. Gelfand I.M., Dorfman I.Ya., Schouten bracket and Hamiltonian operators, Funktsional. Anal. i Prilozhen. 14 (1980), no. 3, 71-74 (English transl.: Funct. Anal. Appl. 14 (1980), no. 3, 223-226).
  41. Gerstenhaber M., The cohomology structure of an associative ring, Ann. of Math. (2) 78 (1963), 267-288.
  42. Gerstenhaber M., Giaquinto A., Boundary solutions of the classical Yang-Baxter equation, Lett. Math. Phys. 40 (1997), 337-353, q-alg/9609014.
  43. Ginzburg V.L., Grothendieck groups of Poisson vector bundles, J. Symplectic Geom. 1 (2001), 121-169, math.DG/0009124.
  44. Ginzburg V.L., Golubev A., Holonomy on Poisson manifolds and the modular class, Israel J. Math. 122 (2001), 221-242, math.DG/9812153.
  45. Grabowski J., Marmo G., Michor P., Homology and modular classes of Lie algebroids, Ann. Inst. Fourier (Grenoble) 56 (2006), 69-83, math.DG/0310072.
  46. Grabowski J., Marmo G., Perelomov A. M., Poisson structures: towards a classification, Modern Phys. Lett. A 8 (1993), 1719-1733.
  47. Grabowski J., Urbanski P., Lie algebroids and Poisson-Nijenhuis structures, Rep. Math. Phys. 40 (1997), 195-208, dg-ga/9710007.
  48. Gualtieri M., Generalized complex geometry, math.DG/0703298.
  49. Higgins P.J., Mackenzie K., Algebraic constructions in the category of Lie algebroids, J. Algebra 129 (1990), 194-230.
  50. Hirota Y., Morita invariant properties of twisted Poisson manifolds, Lett. Math. Phys. 81 (2007), 185-195.
  51. Hitchin N., Generalized Calabi-Yau manifolds, Q. J. Math. 54 (2003), 281-308, math.DG/0209099.
  52. Hodges T., Yakimov M., Triangular Poisson structures on Lie groups and symplectic reduction, in Noncommutative Geometry and Representation Theory in Mathematical Physics, Editors J. Fuchs et al., Contemp. Math. 391 (2005), 123-134, math.SG/0412082.
  53. Huebschmann J., Poisson cohomology and quantization, J. Reine Angew. Math. 408 (1990), 57-113.
  54. Huebschmann J., Lie-Rinehart algebras, Gerstenhaber algebras, and Batalin-Vilkovisky algebras, Ann. Inst. Fourier (Grenoble) 48 (1998), 425-440, dg-ga/9704005.
  55. Huebschmann J., Duality for Lie-Rinehart algebras and the modular class, J. Reine Angew. Math. 510 (1999), 103-159, dg-ga/9702008.
  56. Ibáñez R., de León M., López B., Marrero J.C., Padrón E., Duality and modular class of a Nambu-Poisson structure, J. Phys. A: Math. Gen. 34 (2001), 3623-3650, math.SG/0004065.
  57. Ibáñez R., de León M., Marrero J.C., Padrón E., Leibniz algebroid associated with a Nambu-Poisson structure, J. Phys. A: Math. Gen. 32 (1999), 8129-8144, math-ph/9906027.
  58. Ibáñez R., Lopez B., Marrero J.C., Padrón E., Matched pairs of Leibniz algebroids, Nambu-Jacobi structures and modular class, C. R. Acad. Sci. Paris Sér. I Math. 333 (2001), 861-866.
  59. Ibort A., Martínez Torres D., A new construction of Poisson manifolds, J. Symplectic Geom. 2 (2003), 83-107.
  60. Iglesias D., Lopez B., Marrero J.C., Padrón E., Triangular generalized Lie bialgebroids: homology and cohomology theories, in Lie Algebroids and Related Topics in differential Geometry (2000, Warsaw), Banach Center Publ., Vol. 54, Polish Acad. Sci., Warsaw, 2001, 111-133.
  61. Karasëv M.V., Analogues of objects of the theory of Lie groups for nonlinear Poisson brackets, Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), no. 3, 508-538 (English transl.: Math. USSR Izv. 28 (1987), 497-527).
  62. Klimcík C., Strobl T., WZW-Poisson manifolds, J. Geom. Phys. 43 (2002), 341-344, math.SG/0104189.
  63. Kosmann-Schwarzbach Y., Jacobian quasi-bialgebras and quasi-Poisson Lie groups, in Mathematical Aspects of Classical Field Theory (1991, Seattle, WA), Editors M.J. Gotay, J.E. Marsden and V. Moncrief, Contemp. Math. 132 (1992), 459-489.
  64. Kosmann-Schwarzbach Y., Exact Gerstenhaber algebras and Lie bialgebroids, Acta Appl. Math. 41 (1995), 153-165.
  65. Kosmann-Schwarzbach Y., The Lie bialgebroid of a Poisson-Nijenhuis manifold, Lett. Math. Phys. 38 (1996), 421-428.
  66. Kosmann-Schwarzbach Y., From Poisson algebras to Gerstenhaber algebras, Ann. Inst. Fourier (Grenoble) 46 (1996), 1243-1274.
  67. Kosmann-Schwarzbach Y., Lie bialgebras, Poisson Lie groups and dressing transformations, in Integrability of nonlinear systems (1996, Pondicherry), Lecture Notes in Phys., Vol. 495, Springer, Berlin, 1997, 104-170.
  68. Kosmann-Schwarzbach Y., Modular vector fields and Batalin-Vilkovisky algebras, in Poisson Geometry (1998, Warsaw), Editors J. Grabowski and P. Urbanski, Banach Center Publ., Vol. 51, Polish Acad. Sci., Warsaw, 2000, 109-129.
  69. Kosmann-Schwarzbach Y., Derived brackets, Lett. Math. Phys. 69 (2004), 61-87, math.DG/0312524.
  70. Kosmann-Schwarzbach Y., Quasi, twisted, and all that in Poisson geometry and Lie algebroid theory, in The Breadth of Symplectic and Poisson Geometry, Editors J.E. Marsden and T. Ratiu, Progr. Math., Vol. 232, Birkhäuser, Boston, MA, 2005, 363-389, math.SG/0310359.
  71. Kosmann-Schwarzbach Y., Poisson and symplectic functions in Lie algebroid theory, arXiv:0711.2043.
  72. Kosmann-Schwarzbach Y., Laurent-Gengoux C., The modular class of a twisted Poisson structure, Trav. Math. 16 (2005), 315-339, math.SG/0505663.
  73. Kosmann-Schwarzbach Y., Laurent-Gengoux C., Weinstein A., Modular classes of Lie algebroid morphisms, arXiv:0712.3021.
  74. Kosmann-Schwarzbach Y., Magri F., Poisson-Nijenhuis structures, Ann. Inst. H. Poincaré Phys. Théor. 53 (1990), 35-81.
  75. Kosmann-Schwarzbach Y., Magri F., On the modular classes of Poisson-Nijenhuis manifolds, math.SG/0611202.
  76. Kosmann-Schwarzbach Y., Monterde J., Divergence operators and odd Poisson brackets, Ann. Inst. Fourier (Grenoble) 52 (2002), 419-456, math.QA/0002209.
  77. Kosmann-Schwarzbach Y., Weinstein A., Relative modular classes of Lie algebroids, C. R. Math. Acad. Sci. Paris 341 (2005), 509-514, math.DG/0508515.
  78. Kosmann-Schwarzbach Y., Yakimov M., Modular classes of regular twisted Poisson structures on Lie algebroids, Lett. Math. Phys. 80 (2007), 183-197, math.SG/0701209.
  79. Kostant B., Sternberg S., Symplectic reduction, BRS cohomology, and infinite-dimensional Clifford algebras, Ann. Physics 176 (1987), 49-113.
  80. Koszul J.-L., Crochet de Schouten-Nijenhuis et cohomologie, in The Mathematical Heritage of Élie Cartan (1984, Lyon), Astérisque (1985), 257-271.
  81. Kotov A., Strobl T., Characteristic classes associated to Q-bundles, arXiv:0711.4106.
  82. Kubarski J., The Weil algebra and the secondary characteristic homomorphism of regular Lie algebroids, in Lie Algebroids and Related Topics in Differential Geometry (2000, Warsaw), Banach Center Publ., Vol. 54, Polish Acad. Sci., Warsaw, 2001, 135-173.
  83. Launois S., Richard L., Twisted Poincaré duality for some quadratic Poisson algebras, Lett. Math. Phys. 79 (2007), 161-174, math.KT/0609390.
  84. Laurent-Gengoux C., Stiénon M., Xu P., Holomorphic Poisson structures and groupoids, arXiv:0707.4253.
  85. Laurent-Gengoux C., Tu J.-L., Xu P., Chern-Weil map for principal bundles over groupoids, Math. Z. 255 (2007), 451-491, math.DG/0401420.
  86. Lecomte P., Roger C., Modules et cohomologies des bigèbres de Lie, C. R. Acad. Sci. Paris Sér. I Math. 310 (1990), 405-410, Erratum, 893-894.
  87. Libermann P., Marle C.-M., Symplectic geometry and analytical mechanics, Mathematics and Its Applications, Vol. 35, D. Reidel Publishing Co., Dordrecht, 1987.
  88. Lichnerowicz A., Les variétés de Poisson et leurs algèbres de Lie associées, J. Differential Geom. 12 (1977), 253-300.
  89. Liu Z.J., Xu P., On quadratic Poisson structures, Lett. Math. Phys. 26 (1992), 33-42.
  90. Liu Z.J., Xu P., Exact Lie bialgebroids and Poisson groupoids, Geom. Funct. Anal. 6 (1996), 138-145.
  91. Lu J.-H., A note on Poisson homogeneous spaces, arXiv:0706.1337.
  92. Lyakhovich S.L., Sharapov A.A., Characteristic classes of gauge systems, Nuclear Phys. B 703 (2004), 419-453.
  93. Mackenzie K., Lie groupoids and Lie algebroids in differential geometry, London Mathematical Society Lecture Note Series, Vol. 124, Cambridge University Press, Cambridge, 1987.
  94. Mackenzie K., General theory of Lie groupoids and Lie algebroids, London Mathematical Society Lecture Note Series, Vol. 213, Cambridge University Press, Cambridge, 2005.
  95. Mackenzie K., Xu P., Lie bialgebroids and Poisson groupoids, Duke Math. J. 73 (1994), 415-452.
  96. Magri F., Morosi C., A geometrical characterization of integrable Hamiltonian systems through the theory of Poisson-Nijenhuis manifolds, Quaderno, Vol. 19, University of Milano, 1984.
  97. Meinrenken E., Lectures on pure spinors and moment maps, math.DG/0609319.
  98. Menichi L., Batalin-Vilkovisky algebra structures on Hochschild cohomology, arXiv:0711.1946.
  99. Mikami K., Godbillon-Vey classes of symplectic foliations, Pacific J. Math. 194 (2000), 165-174.
  100. Mitric G., Vaisman I., Poisson structures on tangent bundles, Differential Geom. Appl. 18 (2003), 207-228, math.DG/0108130.
  101. Moerdijk I., Mrcun J., Introduction to foliations and Lie groupoids, Cambridge Studies in Advanced Mathematics, Vol. 91, Cambridge University Press, Cambridge, 2003.
  102. Monterde J., Vallejo J. A., A modular class of even symplectic manifolds, Teoret. Mat. Fiz. 132 (2002), 50-59 (English transl.: Theoret. and Math. Phys. 132 (2002), 934-941).
  103. Neumaier N., Waldmann S., Deformation quantization of Poisson structures associated to Lie algebroids, arXiv:0708.0516.
  104. Palais R., The cohomology of Lie rings, in Proc. Sympos. Pure Math., Vol. 3, Amer. Math. Soc., Providence, RI, 1961, 130-137.
  105. Park J.-S., Topological open p-branes, in Symplectic Geometry and Mirror Symmetry (2000, Seoul), World Sci. Publ., River Edge, NJ, 2001, 311-384.
  106. Pestun V., Topological strings in generalized complex space, Adv. Theor. Math. Phys. 11 (2007), 399-450, hep-th/0603145.
  107. Polishchuk A., Algebraic geometry of Poisson brackets, in Algebraic Geometry 7, J. Math. Sci. (New York) 84 (1997), 1413-1444.
  108. Pradines J., Théorie de Lie pour les groupoïdes différentiables. Calcul différentiel dans la catégorie des groupoïdes infinitésimaux, C. R. Acad. Sci. Paris Sér. A-B 264 (1967), A245-A248.
  109. Radko O., A classification of topologically stable Poisson structures on a compact oriented surface, J. Symplectic Geom. 1 (2002), 523-542, math.SG/0110304.
  110. Radko O., Toward a classification of Poisson structures on surfaces, in Quantization, Poisson Brackets and Beyond (2001, Manchester), Editor T. Voronov, Contemp. Math. 315 (2002), 81-88.
  111. Roytenberg D., Quasi-Lie bialgebroids and twisted Poisson manifolds, Lett. Math. Phys. 61 (2002), 123-137, math.QA/0112152.
  112. Schwarz A., Geometry of Batalin-Vilkovisky quantization, Comm. Math. Phys. 155 (1993), 249-260, hep-th/9205088.
  113. Severa P., Weinstein A., Poisson geometry with a 3-form background, in Noncommutative Geometry and String Theory, Editors Y. Maeda and S. Watamura, Progr. Theoret. Phys. Suppl. 144 (2001), 145-154.
  114. Stolin A., On rational solutions of Yang-Baxter equation for sl(n), Math. Scand. 69 (1991), 57-80.
  115. Terashima Y., On Poisson functions, J. Symplectic Geom., to appear.
  116. Tseng H.-H., Zhu C., Integrating Lie algebroids via stacks, Compos. Math. 142 (2006), 251-270, math.DG/0405003.
  117. Vaintrob A.Yu., Lie algebroids and homological vector fields, Usp. Mat. Nauk 52 (1997), 161-162 (English transl.: Russ. Math. Surv. 52 (1997), 428-429).
  118. Vaisman I., Lectures on the geometry of Poisson manifolds, Progr. Math., Vol. 118, Birkhäuser, Basel, 1994.
  119. Vaisman I., The Poisson-Nijenhuis manifolds revisited, Rend. Sem. Mat. Univ. Politec. Torino 52 (1994), 377-394.
  120. Vaisman I., A lecture on Poisson-Nijenhuis structures, in Integrable Systems and Foliations (1995, Montpellier), Editors C. Albert, R. Brouzet and J.-P. Dufour, Progr. Math., Vol. 145, Birkhäuser, Boston, MA, 1997, 169-185.
  121. Vaisman I., The BV-algebra of a Jacobi manifold, Ann. Polon. Math. 73 (2000), 275-290, math.DG/9904112.
  122. Voronov T., Graded manifolds and Drinfeld doubles for Lie bialgebroids, in Quantization, Poisson Brackets and Beyond (2001, Manchester), Editor T. Voronov, Contemp. Math. 315 (2002), 131-168, math.DG/0105237.
  123. Wade A., On some properties of Leibniz algebroids, in Infinite Dimensional Lie Groups in Geometry and Representation Theory (2000, Washington, DC), World Sci. Publ., River Edge, NJ, 2002, 65-78.
  124. Weinstein A., The modular automorphism group of a Poisson manifold, J. Geom. Phys. 23 (1997), 379-394.
  125. Xu P., Gerstenhaber algebras and BV-algebras in Poisson geometry, Comm. Math. Phys. 200 (1999), 545-560, dg-ga/9703001.
  126. Xu P., Dirac submanifolds and Poisson involutions, Ann. Sci. École Norm. Sup. (4) 36 (2003), 403-430, math.SG/0110326.
  127. Zabzine M., Lectures on generalized complex geometry and supersymmetry, Arch. Math. (Brno) 42 (2006), suppl., 119-146, hep-th/0605148.

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