The Z2-graded Schouten-Nijenhuis bracket and generalized super-Poisson structures

Journal of Mathematical Physics

Volumn 38, Issue 7, 1997, Pages 3735-3749

  De Azcarraga J A ^a,b   Izquierdo, J M ^b   Perelomov, A M ^b,c,d   Perez Bueno J C ^a,b  

^a UNIVERSITY OF CAMBRIDGE   (United Kingdom)

^b INSTITUTO DE FÍSICA CORPUSCULAR IFIC   (Spain)

^c INSTITUTE FOR THEORETICAL AND EXPERIMENTAL PHYSICS   (Russian Federation)

^d UNIVERSITY OF ZARAGOZA   (Spain)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0039121905     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.532065     Document Type: Article

References (46)

1. J.A. de Azcárraga, A.M. Perelomov, and J.C. Pérez Bueno, "New generalized Poisson structures," J. Phys. A 29, L151-L157 (1996); "The Schouten-Nijenhuis bracket, cohomology and generalized Poisson structures," J. Phys. A 29, 7993-8009 (1996).
2. J.A. de Azcárraga, A.M. Perelomov, and J.C. Pérez Bueno, "New generalized Poisson structures," J. Phys. A 29, L151-L157 (1996); "The Schouten-Nijenhuis bracket, cohomology and generalized Poisson structures," J. Phys. A 29, 7993-8009 (1996).
3. Y. Nambu, "Generalized Hamiltonian dynamics," Phys. Rev. D 7, 2405-2412 (1973).
4. F. Bayen and M. Flato, "Remarks concerning Nambu's generalized mechanics," Phys. Rev. D 11, 3049-3053 (1975).
5. N. Mukunda and E. Sudarshan, "Relation between Nambu and Hamiltonian mechanics," Phys. Rev. D 13, 2846-2850 (1976).
6. L. Takhtajan, "On foundations of the generalized Nambu mechanics," Commun. Math. Phys. 160, 295-315 (1994).
7. J.A. Schouten, "Über differentialkonkomitanten zweier kontravarianter Grössen,̈ Proc. Kon. Ned. Akad. Wet. Amsterdam 43, 449-452 (1940).
8. A. Nijenhuis, "Jacobi-type identities for bilinear differential concomitants of certain tensor fields," Indag. Math. 17, 390-403 (1955).
9. A. Lichnerowicz, "Les variétés de Poisson et leurs algèbres de Lie associées," J. Diff. Geom. 12, 253-300 (1977).
10. W.M. Tulczyjew, "Poisson brackets and canonical manifolds," Bull. Acad. Pol. Sci. Math. Astron. 22, 931-934 (1974).
11. D.A. Leites, "New superalgebras and mechanics," Sov. Math. Dokl. 18, 1277-1280 (1977).
12. B. Kupershmidt, "Odd and even Poisson brackets in dynamical systems," Lett. Math. Phys. 9, 323-330 (1985).
13. J. Monterde, "A characterization of graded symplectic structures," Diff. J. Geom. Appl. 2, 81-97 (1992).
14. J. Grabowski, "Graded extensions of Poisson brackets," talk at the Goslar International Colloquium on Group Theoretical Methods in Physics, July 1996.
15. I.S. Krasil'shchik, "Supercanonical algebras and Schouten bracket," Mat. Zametki 49, 70-76 (1991) [Math. Notes 49, 50-54 (1991)].
16. I.S. Krasil'shchik, "Supercanonical algebras and Schouten bracket," Mat. Zametki 49, 70-76 (1991) [Math. Notes 49, 50-54 (1991)].
17. F. Cantrijn, and A. Ibort, "Introduction to Poisson supermanifolds," Diff. Geom. Appl. 1, 133-152 (1991).
18. A. Rogers, "A global theory of supermanifolds," J. Math. Phys. 21, 1352-1365 (1980).
19. B. De Witt, Supermanifolds, 2nd ed. (Cambridge University Press, Cambridge, 1992).
20. F.A. Berezin, Introduction to Superanalysis, (Reidel, Dordrecht, 1987).
21. B. Kostant, "Graded manifolds, graded Lie theory and prequantization," in Lecture Notes in Mathematics (Springer-Verlag, New York, 1977), Vol. 570, p. 177-306.
22. M. Batchelor, "Two approaches to supermanifolds," Trans. Am. Math. Soc. 258, 257-270 (1980).
23. B. Kostant and S. Sternberg, "Symplectic reduction, BRST cohomology and infinite dimensional Clifford algebras," Ann. Phys. 176, 49-113 (1987).
24. A. Frölicher and A. Nijenhuis, "Theory of vector-valued differential forms," Indag. Math. 18, 338-350; 351-359 (1956).
25. Y. Kosmann-Schwarzbach, "From Poisson algebras to Gerstenhaber algebras," Ann. Inst. Fourier 46, 1243-1274 (1996).
26. P.A.B. Lecomte, P.W. Michor and H. Schicketanz, "The multigraded Nijenhuis-Richardson algebra, its universal property and its applications," J. Pure Appl. Alg. 77, 87-102 (1992).
27. C. Chevalley and S. Eilenberg, "Cohomology theory of Lie groups and Lie algebras," Trans. Am. Math. Soc. 63, 85-124 (1948).
28. J.A. de Azcárraga and J.M. Izquierdo, Lie groups, Lie Algebras, Cohomology and Some Applications in Physics (Cambridge University Press, Cambridge, 1995).
29. V.G. Kac, "Lie superalgebras," Adv. Math. 26, 8-96 (1977); "A sketch of lie superalgebra theory," Commun. Math. Phys. 53, 31-64 (1977).
30. V.G. Kac, "Lie superalgebras," Adv. Math. 26, 8-96 (1977); "A sketch of lie superalgebra theory," Commun. Math. Phys. 53, 31-64 (1977).
31. M. Scheunert, The Theory of Lie Superalgebras (Springer-Verlag, New York, 1979).
32. C. Buttin, "Les dérivations des champs de tenseurs et l'invariant différentiel de Schouten," C.R. Acad. Sci. Paris 269A, 87-89 (1969).
33. I.A. Batalin and G.A. Vilkovisky, "Quantization of gauge theories with linearly dependent generators," Phys. Rev. D 28, 2567 (1984) [Errata: 30, 508 (1984)]. For a review see J. Gomis, J. Paris, and S. Samuel, "Antibrackets, antifields and field quantization," Phys. Rep. 259, 1-145 (1995).
34. I.A. Batalin and G.A. Vilkovisky, "Quantization of gauge theories with linearly dependent generators," Phys. Rev. D 28, 2567 (1984) [Errata: 30, 508 (1984)]. For a review see J. Gomis, J. Paris, and S. Samuel, "Antibrackets, antifields and field quantization," Phys. Rep. 259, 1-145 (1995).
35. E. Witten, "A note on the antibracket formalism," Mod. Phys. Lett. A 5, 487-494 (1990).
36. B.H. Lian and G.J. Zuckerman, "New perspectives on the BRST-algebraic structure of string theory," Commun. Math. Phys. 154, 613-646 (1993).
37. B. Zwiebach, "Closed string theory: quantum action and the Batalin-Vilkovisky master equation," Nucl. Phys. B 390, 33-152 (1993).
38. A. Schwarz, "Geometry of Batalin-Vilkovisky quantization," Commun. Math. Phys. 155, 249-260 (1993); M. Alexandrov, M. Kontsevich, A. Schwarz, and O. Zaboronsky, "The geometry of the master equation and topological quantum field theory," Int. J. Mod. Phys. A 12, 1405-1430 (1997).
39. A. Schwarz, "Geometry of Batalin-Vilkovisky quantization," Commun. Math. Phys. 155, 249-260 (1993); M. Alexandrov, M. Kontsevich, A. Schwarz, and O. Zaboronsky, "The geometry of the master equation and topological quantum field theory," Int. J. Mod. Phys. A 12, 1405-1430 (1997).
40. J.A. de Azcárraga and J.C. Pérez Bueno, "Higher-order simple Lie algebras," Commun. Math. Phys. 184, 669-681 (1997).
41. P.D. Jarvis and H.S. Green, "Casimir invariants and characteristic identities for generators of the general linear, special linear and orthosymplectic graded Lie algebras," J. Math. Phys. 20, 2115-2122 (1979).
42. A. N. Sergeev, C. R. Acad. Bulgare Sci. 35, 573 (1982); "Analogue of the classical invariant theory for Lie superalgebras," Funct. Anal. Appl. 26, 223-225 (1992).
43. A. N. Sergeev, C. R. Acad. Bulgare Sci. 35, 573 (1982); "Analogue of the classical invariant theory for Lie superalgebras," Funct. Anal. Appl. 26, 223-225 (1992).
44. V. G. Kac, Proc. Natl. Acad. Sci. USA 81, 645 (1984).
45. M. Scheunert, "Casimir elements of Lie superalgebras," in Differential Geometric Methods in Mathematical Physics, edited by S. Sternberg (Dordrecht, Reidel, 1984); "Invariant supersymmetric multilinear forms and the Casimir elements of P-type Lie superalgebras," J. Math. Phys. 28, 1180-1191 (1987).
46. M. Scheunert, "Casimir elements of Lie superalgebras," in Differential Geometric Methods in Mathematical Physics, edited by S. Sternberg (Dordrecht, Reidel, 1984); "Invariant supersymmetric multilinear forms and the Casimir elements of P-type Lie superalgebras," J. Math. Phys. 28, 1180-1191 (1987).