On définit les opérateurs de divergence sur les algèbres graduées et l’on montre qu’étant donné un crochet de Poisson impair sur l’algèbre, l’opérateur qui associe à un élément la divergence de la dérivation hamiltonienne qu’il définit est un générateur du crochet. C’est le “laplacien impair”, de la quantification de Batalin-Vilkovisky. On étudie alors les générateurs des crochets de Poisson impairs sur les supervariétés, où l’opérateur de divergence peut être défini soit à l’aide d’un volume bérézinien, soit à l’aide d’une connexion graduée. Comme exemples, on trouve des générateurs du crochet de Schouten des multivecteurs sur une variété (la supervariété étant le fibré cotangent, où les coordonnées sur les fibres sont impaires), et ceux du crochet de Koszul-Schouten des formes différentielles sur une variété de Poisson (la supervariété étant le fibré tangent, avec des coordonnées impaires sur les fibres).
We define the divergence operators on a graded algebra, and we show that, given an odd Poisson bracket on the algebra, the operator that maps an element to the divergence of the hamiltonian derivation that it defines is a generator of the bracket. This is the “odd laplacian”, , of Batalin-Vilkovisky quantization. We then study the generators of odd Poisson brackets on supermanifolds, where divergences of graded vector fields can be defined either in terms of berezinian volumes or of graded connections. Examples include generators of the Schouten bracket of multivectors on a manifold (the supermanifold being the cotangent bundle where the coordinates in the fibres are odd) and generators of the Koszul-Schouten bracket of forms on a Poisson manifold (the supermanifold being the tangent bundle, with odd coordinates on the fibres).
@article{AIF_2002__52_2_419_0,
author = {Kosmann-Schwarzbach, Yvette and Monterde, Juan},
title = {Divergence operators and odd Poisson brackets},
journal = {Annales de l'Institut Fourier},
volume = {52},
year = {2002},
pages = {419-456},
doi = {10.5802/aif.1892},
mrnumber = {1906481},
zbl = {1054.53094},
language = {en},
url = {http://dml.mathdoc.fr/item/AIF_2002__52_2_419_0}
}
Kosmann-Schwarzbach, Yvette; Monterde, Juan. Divergence operators and odd Poisson brackets. Annales de l'Institut Fourier, Tome 52 (2002) pp. 419-456. doi : 10.5802/aif.1892. http://gdmltest.u-ga.fr/item/AIF_2002__52_2_419_0/
[1] The geometry of the master equation and topological quantum field theory, Int. J. Mod. Phys, Tome A12 (1997), pp. 1405-1429 | MR 1432574 | Zbl 01712530
[2] Gauge algebra and quantization, Phys. Lett., Tome B 102 (1981), pp. 27-31 | MR 616572
[3] Closure of the gauge algebra, generalized Lie equations and Feynman rules, Nuclear Physics, Tome B 234 (1984), pp. 106-124 | MR 736479
[4] Introduction to Superanalysis, D. Reidel (1987)
[5] Graded Poisson structures on the algebra of differential forms, Comment. Math. Helv., Tome 70 (1995), pp. 383-402 | Article | MR 1340100 | Zbl 0844.58025
[6] Graded Jacobi operators on the algebra of differential forms, Compositio Math., Tome 106 (1997), pp. 43-59 | Article | MR 1446150 | Zbl 0874.58017
[7] Quantum Fields and Strings: A Course for Mathematicians, Amer. Math. Soc., Tome vol. 1, part 1 (1999)
[8] Supermanifolds, Cambridge Univ. Press (1984)
[9] Theory of vector-valued differential forms, part I, Indag. Math, Tome 18 (1956), pp. 338-359 | MR 82554 | Zbl 0079.37502
[10] Batalin-Vilkovisky algebras and two-dimensional topological field theories, Commun. Math. Phys., Tome 159 (1994), pp. 265-285 | Article | MR 1256989 | Zbl 0807.17026
[11] Developing the covariant Batalin-Vilkovisky approach to string theory, Ann. Phys., Tome 229 (1994), pp. 177-216 | Article | MR 1257465 | Zbl 0784.53054
[12] Construction intrinsèque du faisceau de Berezin d'une variété graduée, Comptes Rendus Acad. Sci. Paris, Sér. I Math, Tome 301 (1985), pp. 915-918 | MR 829061 | Zbl 0592.58042
[13] Variational berezinian problems and their relationship with graded variational problems, Diff. Geometric Methods in Math. Phys. (Salamanca 1985), Springer-Verlag (Lect. Notes Math.) Tome 1251 (1987), pp. 137-149 | Zbl 0627.58020
[14] Poisson cohomology and quantization, J. für die reine und angew. Math., Tome 408 (1990), pp. 57-113 | Article | MR 1058984 | Zbl 0699.53037
[15] Lie-Rinehart algebras, Gerstenhaber algebras, and Batalin-Vilkovisky algebras, Ann. Inst. Fourier, Tome 48 (1998) no. 2, pp. 425-440 | Article | Numdam | MR 1625610 | Zbl 0973.17027
[16] Duality for Lie-Rinehart algebras and the modular class, J. für die reine und angew. Math., Tome 510 (1999), pp. 103-159 | Article | MR 1696093 | Zbl 1034.53083
[17] Geometry of superspace with even and odd brackets, J. Math. Phys., Tome 32 (1991), pp. 1934-1937 | Article | MR 1112728 | Zbl 0737.58063
[18] Batalin-Vilkovisky formalism and odd symplectic geometry, Geometry and integrable models (Dubna 1994), World Sci. Publ. (1996), pp. 144-181
[19] On the geometry of the Batalin-Vilkovisky formalism, Mod. Phys. Lett., Tome A 8 (1993), pp. 2377-2385 | MR 1234886 | Zbl 1021.81948
[20] From Poisson algebras to Gerstenhaber algebras, Ann. Inst. Fourier, Tome 46 (1996) no. 5, pp. 1243-1274 | Article | Numdam | MR 1427124 | Zbl 0858.17027
[21] Modular vector fields and Batalin-Vilkovisky algebras, Banach Center Publications, Tome 51 (2000), pp. 109-129 | MR 1764439 | Zbl 1018.17020
[22] Poisson-Nijenhuis structures, Ann. Inst. Henri Poincaré, Tome A53 (1990), pp. 35-81 | Numdam | MR 1077465 | Zbl 0707.58048
[23] Graded manifolds, graded Lie theory and prequantization, Proc. Conf. Diff. Geom. Methods in Math. Phys. (Bonn 1975) (Lecture Notes Math.) Tome 570 (1977), pp. 177-306 | Zbl 0358.53024
[24] Crochet de Schouten-Nijenhuis et cohomologie, Élie Cartan et les mathématiques d'aujourd'hui, Soc. Math. Fr. (Astérisque, hors série) (1985), pp. 257-271 | Zbl 0615.58029
[25] Schouten brackets and canonical algebras, Springer-Verlag (Lecture Notes Math.) Tome 1334 (1988), pp. 79-110 | Zbl 0661.53059
[26] Supercanonical algebras and Schouten brackets, Mat. Zametki, Tome 49(1) (1991), pp. 70-76 | MR 1101552 | Zbl 0723.58020
[26] Supercanonical algebras and Schouten brackets, Mathematical Notes, Tome 49(1) (1991), pp. 50-54 | MR 1101552 | Zbl 0732.58016
[27] Supermanifold Theory, Karelia Branch of the USSR Acad. of Sci., Petrozavodsk (in Russian). (1983)
[28] Quantization and supermanifolds, Kluwer, The Schrödinger Equation, Tome Supplément 3 in Berezin (1991)
[29] New perspectives on the BRST-algebraic structure of string theory, Commun. Math. Phys, Tome 154 (1993), pp. 613-646 | Article | MR 1224094 | Zbl 0780.17029
[30] Gauge Field Theory and Complex Geometry, Springer-Verlag (1988) | MR 954833 | Zbl 0641.53001
[31] The formalism of left and right connections on supermanifolds, Lectures on Supermanifolds, Geometrical Methods and Conformal Groups, World Sci. Publ., Tome Doebner H.-D., Hennig, J. D.PalevT. D.eds. (1989), pp. 3-13 | Zbl 0824.58007
[32] Integral curves of derivations, Ann. Global Anal. Geom., Tome 6 (1988), pp. 177-189 | Article | MR 982764 | Zbl 0632.58017
[33] The exterior derivative as a Killing vector field, Israel J. Math., Tome 93 (1996), pp. 157-170 | Article | MR 1380639 | Zbl 0853.58010
[34] -modules on supermanifolds, Invent. Math., Tome 71 (1983), pp. 501-512 | MR 695902 | Zbl 0528.32012
[35] Integration on noncompact supermanifolds, Trans. Amer. Math. Soc., Tome 299 (1987), pp. 387-396 | Article | MR 869418 | Zbl 0611.58014
[36] Remarks on formal deformations and Batalin-Vilkovisky algebras (e-print, math.AG/9802006)
[37] Geometry of Batalin-Vilkovisky quantization, Commun. Math. Phys., Tome 155 (1993), pp. 249-260 | Article | MR 1230027 | Zbl 0786.58017
[38] Semi-classical approximation in Batalin-Vilkovisky formalism, Commun. Math. Phys., Tome 158 (1993), pp. 373-396 | Article | MR 1249600 | Zbl 0855.58005
[39] Deformation theory and the Batalin-Vilkovisky master equation, Deformation Theory and Symplectic Geometry (Ascona 1996), Kluwer (1997), pp. 271-284
[40] Lectures on the Geometry of Poisson Manifolds, Birkhäuser (1994) | MR 1269545 | Zbl 0810.53019
[41] Geometric integration theory on supermanifolds, Sov. Sci. Rev. C Math, Tome 9 (1992), pp. 1-138 | MR 1202882 | Zbl 0839.58014
[42] The modular automorphism group of a Poisson manifold, J. Geom. Phys., Tome 23 (1997), pp. 379-394 | Article | MR 1484598 | Zbl 0902.58013
[43] A note on the antibracket formalism, Mod. Phys. Lett., Tome A5 (1990), pp. 487-494 | MR 1049114 | Zbl 1020.81931
[44] Gerstenhaber algebras and BV-algebras in Poisson geometry, Commun. Math. Phys., Tome 200 (1999), pp. 545-560 | Article | MR 1675117 | Zbl 0941.17016