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Yvette Kosmann-Schwarzbach; Juan Monterde
Divergence operators and odd Poisson brackets
(Opérateurs de divergence et crochets de Poisson impairs)
Annales de l'institut Fourier, 52 no. 2 (2002), p. 419-456
Article PDF | Reviews MR 1906481 | Zbl 1054.53094 | 1 citation in Cedram
Class. Math.: 17B70, 17B63, 58A50, 81S10, 53D17
Keywords: graded Lie algebras, Gerstenhaber algebra, Batalin-Vilkovisky algebra, Schouten bracket, supermanifold, berezinian volume, graded connection, Maurer-Cartan equation, quantum master equation
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", $\Delta$, 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).
[1] M. Alexandrov, M. Kontsevich, A. Schwarz & O. Zaboronsky, “The geometry of the master equation and topological quantum field theory”, Int. J. Mod. Phys A12 (1997), p. 1405-1429 MR 1432574 | Zbl 01712530
[2] I. A. Batalin & G. A. Vilkovisky, “Gauge algebra and quantization”, Phys. Lett. B 102 (1981), p. 27-31 MR 616572
[3] I. A. Batalin & G. A. Vilkovisky, “Closure of the gauge algebra, generalized Lie equations and Feynman rules”, Nuclear Physics B 234 (1984), p. 106-124 MR 736479
[4] F. A. Berezin, Introduction to Superanalysis, D. Reidel, 1987
[5] J. V. Beltrán & J. Monterde, “Graded Poisson structures on the algebra of differential forms”, Comment. Math. Helv. 70 (1995), p. 383-402
Article | MR 1340100 | Zbl 0844.58025
[6] J. V. Beltrán, J. Monterde, O. A. Sánchez & - Valenzuela, “Graded Jacobi operators on the algebra of differential forms”, Compositio Math. 106 (1997), p. 43-59 MR 1446150 | Zbl 0874.58017
[7] P. Deligne et al., Quantum Fields and Strings: A Course for Mathematicians, Amer. Math. Soc., 1999
[8] B. De & Witt, Supermanifolds, Cambridge Univ. Press, 1984
[9] A. Frölicher & A. Nijenhuis, “Theory of vector-valued differential forms, part I”, Indag. Math 18 (1956), p. 338-359 MR 82554 | Zbl 0079.37502
[10] E. Getzler, “Batalin-Vilkovisky algebras and two-dimensional topological field theories”, Commun. Math. Phys. 159 (1994), p. 265-285
Article | MR 1256989 | Zbl 0807.17026
[11] H. Hata & B. Zwiebach, “Developing the covariant Batalin-Vilkovisky approach to string theory”, Ann. Phys. 229 (1994), p. 177-216 MR 1257465 | Zbl 0784.53054
[12] D. Hernández, Ruipérez, J. Muñoz & Masqué, “Construction intrinsèque du faisceau de Berezin d'une variété graduée”, Comptes Rendus Acad. Sci. Paris, Sér. I Math 301 (1985), p. 915-918 MR 829061 | Zbl 0592.58042
[13] D. Hernández, Ruipérez, J. Muñoz & Masqué, Variational berezinian problems and their relationship with graded variational problems, Lect. Notes Math., Springer-Verlag, 1987, p. 137-149 Zbl 0627.58020
[14] J. Huebschmann, “Poisson cohomology and quantization”, J. für die reine und angew. Math. 408 (1990), p. 57-113 MR 1058984 | Zbl 0699.53037
[15] J. Huebschmann, “Lie-Rinehart algebras, Gerstenhaber algebras, and Batalin-Vilkovisky algebras”, Ann. Inst. Fourier 48 (1998), p. 425-440
Cedram | MR 1625610 | Zbl 0973.17027
[16] J. Huebschmann, “Duality for Lie-Rinehart algebras and the modular class”, J. für die reine und angew. Math. 510 (1999), p. 103-159 MR 1696093 | Zbl 1034.53083
[17] O. M. Khudaverdian, “Geometry of superspace with even and odd brackets”, J. Math. Phys. 32 (1991), p. 1934-1937 MR 1112728 | Zbl 0737.58063
[18] O. M. Khudaverdian, Batalin-Vilkovisky formalism and odd symplectic geometry, World Sci. Publ., 1996, p. 144-181
[19] O. M. Khudaverdian & A. P. Nersessian, “On the geometry of the Batalin-Vilkovisky formalism”, Mod. Phys. Lett. A 8 (1993), p. 2377-2385 MR 1234886 | Zbl 1021.81948
[20] Y. Kosmann & - Schwarzbach, “From Poisson algebras to Gerstenhaber algebras”, Ann. Inst. Fourier 46 (1996), p. 1243-1274
Cedram | MR 1427124 | Zbl 0858.17027
[21] Y. Kosmann & Schwarzbach, “Modular vector fields and Batalin-Vilkovisky algebras”, Banach Center Publications 51 (2000), p. 109-129 MR 1764439 | Zbl 1018.17020
[22] Y. Kosmann, - Schwarzbach & F. Magri, “Poisson-Nijenhuis structures”, Ann. Inst. Henri Poincaré A53 (1990), p. 35-81
Numdam | MR 1077465 | Zbl 0707.58048
[23] B. Kostant, Graded manifolds, graded Lie theory and prequantization, Lecture Notes Math., 1977, p. 177-306 Zbl 0358.53024
[24] J.-L. Koszul, Crochet de Schouten-Nijenhuis et cohomologie, Astérisque, hors série, Soc. Math. Fr., 1985, p. 257-271 Zbl 0615.58029
[25] I. S. Krasil'shchik, Schouten brackets and canonical algebras, Lecture Notes Math., Springer-Verlag, 1988, p. 79-110 Zbl 0661.53059
[26] I. S. Krasil'shchik, “Supercanonical algebras and Schouten brackets”, Mat. Zametki 49(1) (1991), p. 70-76 MR 1101552 | Zbl 0723.58020
[28] D. Leites, Quantization and supermanifolds, The Schrödinger Equation Supplément 3 in Berezin, Kluwer, 1991
[29] B. H. Lian & G. J. Zuckerman, “New perspectives on the BRST-algebraic structure of string theory”, Commun. Math. Phys 154 (1993), p. 613-646
Article | MR 1224094 | Zbl 0780.17029
[30] Y. I. Manin, Gauge Field Theory and Complex Geometry, Springer-Verlag, 1988 MR 954833 | Zbl 0641.53001
[31] Y. I. Manin & I. B. Penkov, The formalism of left and right connections on supermanifolds, World Sci. Publ., 1989, p. 3-13 Zbl 0824.58007
[32] J. Monterde & A. Montesinos, “Integral curves of derivations”, Ann. Global Anal. Geom. 6 (1988), p. 177-189 MR 982764 | Zbl 0632.58017
[33] J. Monterde, O. A. Sánchez & Valenzuela, “The exterior derivative as a Killing vector field”, Israel J. Math. 93 (1996), p. 157-170 MR 1380639 | Zbl 0853.58010
[34] I. B. Penkov, “${\mathcal D}$-modules on supermanifolds”, Invent. Math. 71 (1983), p. 501-512
Article | MR 695902 | Zbl 0528.32012
[35] M. Rothstein, “Integration on noncompact supermanifolds”, Trans. Amer. Math. Soc. 299 (1987), p. 387-396 MR 869418 | Zbl 0611.58014
[36] V. Schechtman, “Remarks on formal deformations and Batalin-Vilkovisky algebras”, e-print, math.AG/9802006
arXiv
[37] A. Schwarz, “Geometry of Batalin-Vilkovisky quantization”, Commun. Math. Phys. 155 (1993), p. 249-260
Article | MR 1230027 | Zbl 0786.58017
[38] A. Schwarz, “Semi-classical approximation in Batalin-Vilkovisky formalism”, Commun. Math. Phys. 158 (1993), p. 373-396
Article | MR 1249600 | Zbl 0855.58005
[39] J. Stasheff, Deformation theory and the Batalin-Vilkovisky master equation, Kluwer, 1997, p. 271-284
[40] I. Vaisman, Lectures on the Geometry of Poisson Manifolds, Birkhäuser, 1994 MR 1269545 | Zbl 0810.53019
[41] T. Voronov, “Geometric integration theory on supermanifolds”, Sov. Sci. Rev. C Math 9 (1992), p. 1-138 MR 1202882 | Zbl 0839.58014
[42] A. Weinstein, “The modular automorphism group of a Poisson manifold”, J. Geom. Phys. 23 (1997), p. 379-394 MR 1484598 | Zbl 0902.58013
[43] E. Witten, “A note on the antibracket formalism”, Mod. Phys. Lett. A5 (1990), p. 487-494 MR 1049114 | Zbl 1020.81931
[44] P. Xu, “Gerstenhaber algebras and BV-algebras in Poisson geometry”, Commun. Math. Phys. 200 (1999), p. 545-560 MR 1675117 | Zbl 0941.17016
[27] D. Leites, “Supermanifold Theory”, Karelia Branch of the USSR Acad. of Sci., Petrozavodsk (in Russian). (1983)
[<L>26</L>] I. S. Krasil'shchik, “Supercanonical algebras and Schouten brackets”, Mathematical Notes 49(1) (1991), p. 50-54 MR 1101552 | Zbl 0732.58016
Yvette Kosmann-Schwarzbach; Juan Monterde
Divergence operators and odd Poisson brackets
(Opérateurs de divergence et crochets de Poisson impairs)
Annales de l'institut Fourier, 52 no. 2 (2002), p. 419-456
Article PDF | Reviews MR 1906481 | Zbl 1054.53094 | 1 citation in Cedram
Class. Math.: 17B70, 17B63, 58A50, 81S10, 53D17
Keywords: graded Lie algebras, Gerstenhaber algebra, Batalin-Vilkovisky algebra, Schouten bracket, supermanifold, berezinian volume, graded connection, Maurer-Cartan equation, quantum master equation
Résumé - Abstract
Bibliography
[2] I. A. Batalin & G. A. Vilkovisky, “Gauge algebra and quantization”, Phys. Lett. B 102 (1981), p. 27-31 MR 616572
[3] I. A. Batalin & G. A. Vilkovisky, “Closure of the gauge algebra, generalized Lie equations and Feynman rules”, Nuclear Physics B 234 (1984), p. 106-124 MR 736479
[4] F. A. Berezin, Introduction to Superanalysis, D. Reidel, 1987
[5] J. V. Beltrán & J. Monterde, “Graded Poisson structures on the algebra of differential forms”, Comment. Math. Helv. 70 (1995), p. 383-402
Article | MR 1340100 | Zbl 0844.58025
[6] J. V. Beltrán, J. Monterde, O. A. Sánchez & - Valenzuela, “Graded Jacobi operators on the algebra of differential forms”, Compositio Math. 106 (1997), p. 43-59 MR 1446150 | Zbl 0874.58017
[7] P. Deligne et al., Quantum Fields and Strings: A Course for Mathematicians, Amer. Math. Soc., 1999
[8] B. De & Witt, Supermanifolds, Cambridge Univ. Press, 1984
[9] A. Frölicher & A. Nijenhuis, “Theory of vector-valued differential forms, part I”, Indag. Math 18 (1956), p. 338-359 MR 82554 | Zbl 0079.37502
[10] E. Getzler, “Batalin-Vilkovisky algebras and two-dimensional topological field theories”, Commun. Math. Phys. 159 (1994), p. 265-285
Article | MR 1256989 | Zbl 0807.17026
[11] H. Hata & B. Zwiebach, “Developing the covariant Batalin-Vilkovisky approach to string theory”, Ann. Phys. 229 (1994), p. 177-216 MR 1257465 | Zbl 0784.53054
[12] D. Hernández, Ruipérez, J. Muñoz & Masqué, “Construction intrinsèque du faisceau de Berezin d'une variété graduée”, Comptes Rendus Acad. Sci. Paris, Sér. I Math 301 (1985), p. 915-918 MR 829061 | Zbl 0592.58042
[13] D. Hernández, Ruipérez, J. Muñoz & Masqué, Variational berezinian problems and their relationship with graded variational problems, Lect. Notes Math., Springer-Verlag, 1987, p. 137-149 Zbl 0627.58020
[14] J. Huebschmann, “Poisson cohomology and quantization”, J. für die reine und angew. Math. 408 (1990), p. 57-113 MR 1058984 | Zbl 0699.53037
[15] J. Huebschmann, “Lie-Rinehart algebras, Gerstenhaber algebras, and Batalin-Vilkovisky algebras”, Ann. Inst. Fourier 48 (1998), p. 425-440
Cedram | MR 1625610 | Zbl 0973.17027
[16] J. Huebschmann, “Duality for Lie-Rinehart algebras and the modular class”, J. für die reine und angew. Math. 510 (1999), p. 103-159 MR 1696093 | Zbl 1034.53083
[17] O. M. Khudaverdian, “Geometry of superspace with even and odd brackets”, J. Math. Phys. 32 (1991), p. 1934-1937 MR 1112728 | Zbl 0737.58063
[18] O. M. Khudaverdian, Batalin-Vilkovisky formalism and odd symplectic geometry, World Sci. Publ., 1996, p. 144-181
[19] O. M. Khudaverdian & A. P. Nersessian, “On the geometry of the Batalin-Vilkovisky formalism”, Mod. Phys. Lett. A 8 (1993), p. 2377-2385 MR 1234886 | Zbl 1021.81948
[20] Y. Kosmann & - Schwarzbach, “From Poisson algebras to Gerstenhaber algebras”, Ann. Inst. Fourier 46 (1996), p. 1243-1274
Cedram | MR 1427124 | Zbl 0858.17027
[21] Y. Kosmann & Schwarzbach, “Modular vector fields and Batalin-Vilkovisky algebras”, Banach Center Publications 51 (2000), p. 109-129 MR 1764439 | Zbl 1018.17020
[22] Y. Kosmann, - Schwarzbach & F. Magri, “Poisson-Nijenhuis structures”, Ann. Inst. Henri Poincaré A53 (1990), p. 35-81
Numdam | MR 1077465 | Zbl 0707.58048
[23] B. Kostant, Graded manifolds, graded Lie theory and prequantization, Lecture Notes Math., 1977, p. 177-306 Zbl 0358.53024
[24] J.-L. Koszul, Crochet de Schouten-Nijenhuis et cohomologie, Astérisque, hors série, Soc. Math. Fr., 1985, p. 257-271 Zbl 0615.58029
[25] I. S. Krasil'shchik, Schouten brackets and canonical algebras, Lecture Notes Math., Springer-Verlag, 1988, p. 79-110 Zbl 0661.53059
[26] I. S. Krasil'shchik, “Supercanonical algebras and Schouten brackets”, Mat. Zametki 49(1) (1991), p. 70-76 MR 1101552 | Zbl 0723.58020
[28] D. Leites, Quantization and supermanifolds, The Schrödinger Equation Supplément 3 in Berezin, Kluwer, 1991
[29] B. H. Lian & G. J. Zuckerman, “New perspectives on the BRST-algebraic structure of string theory”, Commun. Math. Phys 154 (1993), p. 613-646
Article | MR 1224094 | Zbl 0780.17029
[30] Y. I. Manin, Gauge Field Theory and Complex Geometry, Springer-Verlag, 1988 MR 954833 | Zbl 0641.53001
[31] Y. I. Manin & I. B. Penkov, The formalism of left and right connections on supermanifolds, World Sci. Publ., 1989, p. 3-13 Zbl 0824.58007
[32] J. Monterde & A. Montesinos, “Integral curves of derivations”, Ann. Global Anal. Geom. 6 (1988), p. 177-189 MR 982764 | Zbl 0632.58017
[33] J. Monterde, O. A. Sánchez & Valenzuela, “The exterior derivative as a Killing vector field”, Israel J. Math. 93 (1996), p. 157-170 MR 1380639 | Zbl 0853.58010
[34] I. B. Penkov, “${\mathcal D}$-modules on supermanifolds”, Invent. Math. 71 (1983), p. 501-512
Article | MR 695902 | Zbl 0528.32012
[35] M. Rothstein, “Integration on noncompact supermanifolds”, Trans. Amer. Math. Soc. 299 (1987), p. 387-396 MR 869418 | Zbl 0611.58014
[36] V. Schechtman, “Remarks on formal deformations and Batalin-Vilkovisky algebras”, e-print, math.AG/9802006
arXiv
[37] A. Schwarz, “Geometry of Batalin-Vilkovisky quantization”, Commun. Math. Phys. 155 (1993), p. 249-260
Article | MR 1230027 | Zbl 0786.58017
[38] A. Schwarz, “Semi-classical approximation in Batalin-Vilkovisky formalism”, Commun. Math. Phys. 158 (1993), p. 373-396
Article | MR 1249600 | Zbl 0855.58005
[39] J. Stasheff, Deformation theory and the Batalin-Vilkovisky master equation, Kluwer, 1997, p. 271-284
[40] I. Vaisman, Lectures on the Geometry of Poisson Manifolds, Birkhäuser, 1994 MR 1269545 | Zbl 0810.53019
[41] T. Voronov, “Geometric integration theory on supermanifolds”, Sov. Sci. Rev. C Math 9 (1992), p. 1-138 MR 1202882 | Zbl 0839.58014
[42] A. Weinstein, “The modular automorphism group of a Poisson manifold”, J. Geom. Phys. 23 (1997), p. 379-394 MR 1484598 | Zbl 0902.58013
[43] E. Witten, “A note on the antibracket formalism”, Mod. Phys. Lett. A5 (1990), p. 487-494 MR 1049114 | Zbl 1020.81931
[44] P. Xu, “Gerstenhaber algebras and BV-algebras in Poisson geometry”, Commun. Math. Phys. 200 (1999), p. 545-560 MR 1675117 | Zbl 0941.17016
[27] D. Leites, “Supermanifold Theory”, Karelia Branch of the USSR Acad. of Sci., Petrozavodsk (in Russian). (1983)
[<L>26</L>] I. S. Krasil'shchik, “Supercanonical algebras and Schouten brackets”, Mathematical Notes 49(1) (1991), p. 50-54 MR 1101552 | Zbl 0732.58016
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