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Formal deformations of Dirac structures. (English) Zbl 1112.58018

Author’s abstract: In this paper we set-up a general framework for a formal deformation theory of Dirac structures. We give a parameterization of formal deformations in terms of two-forms obeying a cubic equation. The notion of equivalence is discussed in detail. We show that the obstruction for the construction of deformations order by order lies in the third Lie algebroid cohomology of the Dirac structure. However, the classification of inequivalent first order deformations is not given by the second Lie algebroid cohomology but turns out to be more complicated.

MSC:

58H15 Deformations of general structures on manifolds
53D17 Poisson manifolds; Poisson groupoids and algebroids
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[1] Bayen, F.; Flato, M.; Frønsdal, C.; Lichnerowicz, A.; Sternheimer, D., Deformation theory and quantization, Ann. Phys., 111, 61-151 (1978) · Zbl 0377.53025
[2] M. Bordemann, On the deformation quantization of super-Poisson brackets, Preprint (Freiburg FR-THEP-96/8) q-alg/9605038, May 1996; M. Bordemann, On the deformation quantization of super-Poisson brackets, Preprint (Freiburg FR-THEP-96/8) q-alg/9605038, May 1996
[3] Bordemann, M., The deformation quantization of certain super-Poisson brackets and BRST cohomology, (Dito, G.; Sternheimer, D., Conférence Moshé Flato 1999. Quantization, Deformations, and Symmetries. Conférence Moshé Flato 1999. Quantization, Deformations, and Symmetries, Mathematical Physics Studies, no. 22 (2000), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht, Boston, London), 45-68 · Zbl 1004.53067
[4] Bursztyn, H.; Radko, O., Gauge equivalence of Dirac structures and symplectic groupoids, Ann. Inst. Fourier, 53, 309-337 (2003) · Zbl 1026.58019
[5] Cannas da Silva, A.; Weinstein, A., (Geometric Models for Noncommutative Algebras. Geometric Models for Noncommutative Algebras, Berkeley Mathematics Lecture Notes, vol. 10 (1999), AMS) · Zbl 1135.58300
[6] Cariñena, J. F.; Grabowski, J.; Marmo, G., Courant algebroid and Lie bialgebroid contractions, J. Phys. A, 37, 19, 5189-5202 (2004) · Zbl 1058.53022
[7] Courant, T. J., Dirac manifolds, Trans. Amer. Math. Soc., 319, 2, 631-661 (1990) · Zbl 0850.70212
[8] Crainic, M.; Fernandes, R. L., Integrability of Lie brackets, Ann. of Math. (2), 157, 2, 575-620 (2003) · Zbl 1037.22003
[9] M. Crainic, I. Moerdijk, Deformations of Lie brackets: Cohomologial aspects, Preprint math.DG/0403434, 2004; M. Crainic, I. Moerdijk, Deformations of Lie brackets: Cohomologial aspects, Preprint math.DG/0403434, 2004
[10] Dito, G.; Sternheimer, D., Deformation quantization: Genesis, developments and metamorphoses, (Halbout, G., Deformation Quantization. Deformation Quantization, IRMA Lectures in Mathematics and Theoretical Physics, vol. 1 (2002), Walter de Gruyter: Walter de Gruyter Berlin, New York), 9-54 · Zbl 1014.53054
[11] C. Eilks, BRST-Reduktion linearer Zwangsbedingungen im Rahmen der Deformationsquantisierung, Master Thesis, Fakultät für Physik, Albert-Ludwigs-Universität, Freiburg, 2004; C. Eilks, BRST-Reduktion linearer Zwangsbedingungen im Rahmen der Deformationsquantisierung, Master Thesis, Fakultät für Physik, Albert-Ludwigs-Universität, Freiburg, 2004
[12] Fedosov, B. V., Deformation Quantization and Index Theory (1996), Akademie Verlag: Akademie Verlag Berlin · Zbl 0867.58061
[13] Gerstenhaber, M., On the deformation of rings and algebras, Ann. Math., 79, 59-103 (1964) · Zbl 0123.03101
[14] Gerstenhaber, M.; Schack, S. D., Algebraic cohomology and deformation theory, (Hazewinkel, M.; Gerstenhaber, M., Deformation Theory of Algebras and Structures and Applications (1988), Kluwer Academic Press: Kluwer Academic Press Dordrecht), 13-264 · Zbl 0544.18005
[15] M. Gualtieri, Generalized complex geometry, Ph.D. Thesis, St John’s College, University of Oxford, Oxford, 2003, math.DG/0401221; M. Gualtieri, Generalized complex geometry, Ph.D. Thesis, St John’s College, University of Oxford, Oxford, 2003, math.DG/0401221 · Zbl 1235.32020
[16] Gutt, S., Variations on deformation quantization, (Dito, G.; Sternheimer, D., Conférence Moshé Flato 1999. Quantization, Deformations, and Symmetries. Conférence Moshé Flato 1999. Quantization, Deformations, and Symmetries, Mathematical Physics Studies, no. 21 (2000), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht, Boston, London), 217-254 · Zbl 0997.53068
[17] F. Keller, Deformation von Lie-Algebroiden und Dirac-Strukturen, Master Thesis, Fakultät für Mathematik und Physik, Physikalisches Institut, Albert-Ludwigs-Universität, Freiburg, 2004; F. Keller, Deformation von Lie-Algebroiden und Dirac-Strukturen, Master Thesis, Fakultät für Mathematik und Physik, Physikalisches Institut, Albert-Ludwigs-Universität, Freiburg, 2004
[18] Kontsevich, M., Deformation quantization of Poisson manifolds, Lett. Math. Phys., 66, 157-216 (2003) · Zbl 1058.53065
[19] Kosmann-Schwarzbach, Y., From Poisson algebras to Gerstenhaber algebras, Ann. Inst. Fourier (Grenoble), 46, 5, 1243-1274 (1996) · Zbl 0858.17027
[20] Kosmann-Schwarzbach, Y., Derived brackets, Lett. Math. Phys., 69, 61-87 (2004) · Zbl 1055.17016
[21] Kosmann-Schwarzbach, Y., Quasi, twisted, and all that …in Poisson geometry and Lie algebroid theory, (Marsden, J. E.; Ratiu, T. S., The Breadth of Symplectic and Poisson Geometry. The Breadth of Symplectic and Poisson Geometry, Progress in Mathematics, vol. 232 (2005), Birkhäuser Boston Inc.: Birkhäuser Boston Inc. Boston, MA), 363-389, Festschrift in honor of Alan Weinstein · Zbl 1079.53126
[22] Kosmann-Schwarzbach, Y.; Magri, F., Poisson Nijenhuis structures, Ann. Inst. H. Poincaré Phys. Théor., 53, 35-81 (1990) · Zbl 0707.58048
[23] Laurent-Gengoux, C.; Ponte, D.; Xu, P., Universal lifting theorem and quasi-Poisson groupoids (2005)
[24] Liu, Z.; Weinstein, A.; Xu, P., Manin triples for Lie bialgebroids, J. Differential Geom., 45, 3, 547-574 (1997) · Zbl 0885.58030
[25] Mackenzie, K. C.H., (General Theory of Lie Groupoids and Lie Algebroids. General Theory of Lie Groupoids and Lie Algebroids, London Mathematical Society Lecture Note Series, vol. 213 (2005), Cambridge University Press: Cambridge University Press Cambridge, UK) · Zbl 1078.58011
[26] Mackenzie, K. C.H.; Xu, P., Lie bialgebroids and Poisson groupoids, Duke Math. J., 73, 2, 415-452 (1994) · Zbl 0844.22005
[27] Marle, C.-M., Differential calculus on a Lie algebroid and Poisson manifolds, (The J.A. Pereira da Silva Birthday Schrift. The J.A. Pereira da Silva Birthday Schrift, Textos Mat. Sér. B, vol. 32 (2002), Univ. Coimbra: Univ. Coimbra Coimbra), 83-149 · Zbl 1031.53114
[28] Rothstein, M., The structure of supersymplectic supermanifolds, (Bartocci, C.; Bruzzo, U.; Cianci, R., Differential Geometric Methods in Theoretical Physics (Rapallo, 1990) (Proceedings of the Nineteenth International Conference held in Rapallo, June 19-24, 1990). Differential Geometric Methods in Theoretical Physics (Rapallo, 1990) (Proceedings of the Nineteenth International Conference held in Rapallo, June 19-24, 1990), Lecture Notes in Physics, vol. 375 (1991), Springer: Springer Berlin), 331-343 · Zbl 0747.58010
[29] D. Roytenberg, Courant Algebroids, derived brackets and even symplectic supermanifolds, Ph.D. Thesis, UC Berkeley, Berkeley, 1999, math.DG/9910078; D. Roytenberg, Courant Algebroids, derived brackets and even symplectic supermanifolds, Ph.D. Thesis, UC Berkeley, Berkeley, 1999, math.DG/9910078
[30] Roytenberg, D., On the structure of graded symplectic supermanifolds and Courant algebroids, (Voronov, T., Quantization, Poisson brackets and beyond (Manchester, 2001). Quantization, Poisson brackets and beyond (Manchester, 2001), Contemporary Mathematics, vol. 315 (2002), American Mathematical Society: American Mathematical Society Providence, RI), 169-185 · Zbl 1036.53057
[31] Roytenberg, D., Quasi-Lie bialgebroids and twisted Poisson manifolds, Lett. Math. Phys., 61, 2, 123-137 (2002) · Zbl 1027.53104
[32] P. Ševera, On deformation quantization of dirac structures, Preprint math.QA/0511403, 2005; P. Ševera, On deformation quantization of dirac structures, Preprint math.QA/0511403, 2005
[33] Ševera, P.; Weinstein, A., Poisson geometry with a 3-form background, (Maeda, Y.; Watamura, S., Noncommutative Geometry and String Theory (Proceedings of the International Workshop on Noncommutative Geometry and String Theory). Noncommutative Geometry and String Theory (Proceedings of the International Workshop on Noncommutative Geometry and String Theory), Prog. Theo. Phys. Suppl., vol. 144 (2001), Yukawa Institute for Theoretical Physics), 145-154 · Zbl 1029.53090
[34] Uchino, K., Remarks on the definition of a Courant algebroid, Lett. Math. Phys., 60, 2, 171-175 (2002) · Zbl 0999.22004
[35] Weinstein, A., Deformation quantization, Astérisque, 227, 389-409 (1995) · Zbl 0854.58026
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