Geodesic
How to Quantize the Antibracket
Teoretičeskaâ i matematičeskaâ fizika, Tome 126 (2001) no. 3, pp. 339-369 Cet article a éte moissonné depuis la source Math-Net.Ru

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We show that in contrast to $\mathfrak{po}(2n|m)$, its quotient modulo center, the Lie superalgebra $\mathfrak{h}(2n|m)$ of Hamiltonian vector fields with polynomial coefficients, has exceptional additional deformations for $(2n|m)=(2|2)$ and only for this superdimension. We relate this result to the complete description of deformations of the antibracket (also called the Schouten or Buttin bracket). It turns out that the space in which the deformed Lie algebra (result of quantizing the Poisson algebra) acts coincides with the simplest space in which the Lie algebra of commutation relations acts. This coincidence is not necessary for Lie superalgebras. 
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D. A. Leites; I. M. Shchepochkina. How to Quantize the Antibracket. Teoretičeskaâ i matematičeskaâ fizika, Tome 126 (2001) no. 3, pp. 339-369. http://geodesic.mathdoc.fr/item/TMF_2001_126_3_a0/

[1] Yu. Yu. Kochetkov, Deformatsii superalgebr Li, Dep. VINITI, No 384–385, VINITI, M., 1985

[2] D. A. Leites, DAN SSSR, 236 (1977), 804–807 | MR | Zbl

[3] B. Kostant, S. Sternberg, Ann. Phys., 176:1 (1987), 49–113 | DOI | MR | Zbl

[4] D. A. Leites, TMF, 58:2 (1984), 229–232 ; D. Leites, “Quantization. Supplement 3”, Schrödinger Equation, eds. F. Berezin, M. Shubin, Kluwer, Dordrecht, 1991, 483–522 | MR | DOI | MR

[5] B. Fedosov, J. Diff. Geom., 40:2 (1994), 213–238 ; Deformation Quantization and Index Theory, Akademie Verlag, Berlin, 1996 ; I. Gelfand, V. Retakh, M. Shubin, Adv. Math., 136:1 (1998), 104–140 | DOI | MR | Zbl | MR | DOI | MR | Zbl

[6] P. Deligne, Selecta Math. (U. S), 1:4 (1995), 667–697 | DOI | MR | Zbl

[7] J. Vey, Comment. Math. Helv., 50 (1975), 421–454 | DOI | MR | Zbl

[8] M. Hazewinkel, M. Gerstenhaber (eds.), Deformation Theory of Algebras and Structures and Appllications, Kluwer, Dordrecht, 1988 | MR

[9] F. A. Berezin, Commun. Math. Phys., 40 (1975), 153–174 ; Ф. А. Березин, Изв. АН СССР. Сер. матем., 38 (1974), 1116–1175 ; 39:2 (1975), 363–402 | DOI | MR | Zbl | MR | Zbl | MR | Zbl

[10] I. Shereshevskii, DAN SSSR, 245:5 (1979), 1057–1060 ; I. A. Shereshevskij, Lett. Math. Phys., 5:5 (1981), 429–435 | MR | DOI | MR | Zbl

[11] M. Kontsevich, Deformation quantization of Poisson manifolds, E-print q-alg/9709040 | MR

[12] M. Bordemann, The deformation quantization of certain super-Poisson brackets and BRST cohomology, E-print math.QA/0003218 | MR

[13] P. Grozman, D. Leites, I. Shchepochkina, Lie superalgebras of string theories, E-print hep-th 9702120 | MR

[14] I. Batalin, I. Tyutin, General local solution to the cyclic Jacobi equation, unpublished

[15] F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, D. Sternheimer, Ann. Phys., 111:1 (1978), 61–110 | DOI | MR | Zbl

[16] A. Dzhumadildaev, Z. Phys. C, 72:3 (1996), 509–517 | DOI | MR | Zbl

[17] D. A. Leites, UMN, 35:1 (1980), 3–57 ; Теория супермногообразий, Карельское Отделение АН СССР, Петрозаводск, 1983 | MR | Zbl

[18] D. B. Fuks, Kogomologii beskonechnomernykh algebr Li, Nauka, M., 1984 | MR | Zbl

[19] Yu. I. Manin, Kalibrovochnye polya i kompleksnaya geometriya, Nauka, M., 1984 | MR

[20] S. Cheng, V. Kac, Adv. Theor. Math. Phys., 2:5 (1998), 1141–1182 | DOI | MR | Zbl

[21] J. Wess, B. Zumino, Nucl. Phys. B, 70 (1974), 39–50 ; J. Wess, “Supersymmetry-supergravity”, Topics in Quantum Field Theory and Gauge Theories, Proc. of VIII Internat. GIFT Sem. on Theoret. Phys. (Salamanca, 1977), Lect. Notes Phys., 77, ed. J. A. de Azcárraga, Springer, Berlin–New York, 1978, 81–125 ; J. Wess, B. Zumino, Phys. Lett. B, 66:4 (1977), 361–364 ; J. Wess, “Supersymmetry/supergravity”, Concepts and Trends in Particle Physics (Schladming, 1986), eds. H. Latal and H. Mitter, Springer, Berlin, 1987, 29–58 ; “Introduction to supersymmetric theories”, Frontiers in Particle Physics '83 (Dubrovnik, 1983), eds. Dj Šijački, N. Bilić, B. Dragović, D. Popović, World Sci. Publishing, Singapore, 1984, 104–131 ; J. Wess, J. Bagger, Supersymmetry and Supergravity, Princeton University Press, Princeton, NJ, 1983 | DOI | MR | DOI | MR | DOI | MR | MR | MR | MR | Zbl

[22] E. Witten, Phys. Lett. B, 77:4–5 (1978), 394–400 ; “Introduction to supersymmetry in particle and nuclear physics”, Proc. of the International School on Supersymmetry (Mexico City, December 14–18, 1981), eds. O. Castanos, A. Frank, L. Urrutia, Plenum, New York, 1984, 53–76 | DOI | MR

[23] L. E. Gendenshtein, I. V. Krive, UFN, 146:4 (1985), 553–590 | DOI | MR

[24] A. Raina, C. R. Acad. Sci. Paris. Sér. I Math., 318:9 (1994), 851–856 | MR | Zbl

[25] M. A. Shubin, Geom. Funct. Anal., 6:2 (1996), 370–409 | DOI | MR | Zbl

[26] J. Bernstein, The Lie superalgebra $\frak{osp}(1\mid 2)$, connections over symplectic manifolds and representations of Poisson algebras. Seminar on Supermanifolds, Reports of Stockholm Univ. 9/1987-13, ed. D. Leites, Stockholm Univ. Press, Stockholm, 1987, p. 1–60; А. В. Шаповалов, Serdica, 7:4 (1981), 337–342 ; Г. С. Шмелев, Serdica, 8:4 (1982), 408–417 ; Функц. анализ и его прилож., 17:1 (1983), 91–92 | MR | Zbl | MR | Zbl | MR | Zbl

[27] G. S. Shmelev, Funkts. analiz i ego prilozh., 17:4 (1983), 94–95 ; 1, 91–92 ; Serdica, 8:4 (1982), 408–417 (1983) ; Матем. сб., 120:3 (1983), 528–539 ; C. R. Acad. Bulg. Sci., 35:3 (1982), 287–290 | MR | Zbl | MR | Zbl | MR | Zbl | MR | MR | Zbl

[28] Yu. Yu. Kochetkov, “Indutsirovannye neprivodimye predstavleniya superalgebr Leitesa”, Voprosy teorii grupp i gomologicheskoi algebry, eds. A. Onischik, Yaroslavskii gos. universitet, Yaroslavl, 1983, 120–123 ; Вопр. теории групп и гомологич. алгебры, 9 (1989), 142–148 | MR | MR | Zbl

[29] F. A. Berezin, UMN, 24:4 (148) (1969), 65–88 | MR

[30] J. Gomis, J. Paris, S. Samuel, Phys. Rep., 259 (1995), 1–191 | DOI | MR

[31] I. Batalin, I. Tyutin, “Generalized field-antifield formalism”, Topics in Statistical and Theoretical Physics, F. A. Berezin memorial volume, Transl. of AMS. Ser. 2, 177, eds. R. Dobrushin et al., AMS, Providence, RI, 1996, 23–43 | MR

[32] D. A. Leites, “Superalgebry Li”, Sovremennye problemy matematiki. Noveishie dostizheniya, 25, ed. R. V. Gamkrelidze, VINITI, M., 1984, 3–49 | MR

[33] Yu. Yu. Kochetkov, UMN, 44:5 (1989), 167–168 | MR | Zbl

[34] Yu. Kotchetkoff, C. R. Acad. Sci. Paris. Ser. I, 299:14 (1984), 643–645 | MR | Zbl

[35] Yu. Yu. Kochetkov, Matem. zametki, 63:3 (1998), 391–401 ; N. van den Hijligenberg, Y. Kotchetkov, G. Post, J. Algebra Comput., 3:1 (1993), 57–77 | DOI | MR | Zbl | DOI | MR

[36] N. W. van den Hijligenberg, Yu. Yu. Kotchetkov, J. Math. Phys., 37:11 (1996), 5858–5868 | DOI | MR | Zbl

[37] N. van den Hijligenberg, Y. Kotchetkov, G. Post, Math. Comput., 64:211 (1995), 1215–1226 ; Ю. Ю. Кочетков, Функц. анализ и его прилож., 28:3 (1994), 77–79 | DOI | MR | Zbl | MR | Zbl

[38] I. M. Schepochkina, Funkts. analiz i ego prilozh., 33:3 (1999), 59–72 ; ; I. Shchepochkina, Represent. Theory (electr. jour.), 3 (1999), 373–415 ; I. Shchepochkina, G. Post, Internat. J. Algebra Comput., 8:4 (1998), 479–495 ; E-print hep-th/9702120E-print physics/9703022 | DOI | MR | DOI | MR | Zbl | DOI | MR | Zbl

[39] D. Leites, I. Shchepochkina, Classification of simple Lie superalgebras of vector fields (to appear)

[40] D. Alekseevskii, D. Leites, I. Schepochkina, C. R. Acad. Bulg. Sci., 34:9 (1980), 1187–1190

[41] V. Kac, Adv. Math., 26 (1977), 8–96 | DOI | MR | Zbl

[42] V. Serganova, Funkts. analiz i ego prilozh., 17:3 (1983), 46–54 ; 19:3 (1985), 75–76 ; “Outer automorphisms and real forms of Kac–Moody superalgebras”, Group Theoretical Methods in Physics, V. 1–3 (Zvenigorod, 1982), eds. M. I. Markov et al., Harwood Academic Publ., Chur, 1985, 639–642 | MR | Zbl | MR | Zbl

[43] G. S. Shmelev, C. R. Acad. Bulgar. Sci., 36:5 (1983), 569–570 | MR | Zbl

[44] A. Perelomov, Integrable Systems of Classical Mechanics and Lie Algebras, V. I, Birkhäuser, Basel, 1990 | MR | Zbl

[45] C. Duval, P. Lecomte, V. Ovsienko, Ann. Inst. Fourier (Grenoble), 49:6 (1999), 1999–2029 ; E-print math.DG/9902032 | DOI | MR | Zbl

[46] P. B. A. Lecomte, V. Yu. Ovsienko, Projectively equivariant symbol calculus, ; Lett. Math. Phys., 49:3 (1999), 173–196 E-print math.DG/9809061 | MR | DOI | Zbl

[47] D. Leites, E. Poletaeva, Math. Scand., 81:1 (1997), 5–19 | DOI | MR | Zbl

[48] P. Grozman, D. Leites, E. Poletaeva, “Defining relations for simple Lie superalgebras of polynomial vector fields”, Supersymmetries and Quantum Symmetries, SQS'99 (27–31 July, 1999), eds. E. Ivanov et al., JINR, Dubna, 2000, 387–396

[49] R. Floreanini, D. Leites, L. Vinet, “On the defining relations of quantum Poisson superalgebras” (to appear)

[50] D. Leites, A. Shapovalov, J. Nonlinear Math. Phys., 7:2 (2000), 120–125 | DOI | MR | Zbl

[51] P. Ya. Grozman, Funkts. analiz i ego prilozh., 14:2 (1980), 58–59 | MR | Zbl

[52] V. Guillemin, Sh. Sternberg, Symplectic Techniques in Physics, Cambridge University Press, Cambridge, 1990 | MR | Zbl

[53] D. B. Fuchs, D. Leites, C. R. Acad. Bulg. Sci., 37:12 (1984), 1595–1596 | MR | Zbl

[54] A. Sergeev, An analog of the classical invariant theory for Lie superalgebras, E-print math.RT/9810113 | MR

[55] P. Deligne et al. (eds.), Quantum Fields and Strings: A Course for Mathematicians, Material from the Special Year on Quantum Field Theory. V. 1, 2 (Institute for Advanced Study, 1996–1997), AMS, Providence, RI; IAS, Princeton, NJ, 1999 | MR

[56] A. Sergeev, Irreducible representations of solvable Lie superalgebras, ; Represent. Theory (electr. jour.), 3 (1999), 435–443 E-print math.RT/9810109 | MR | DOI | Zbl

[57] J. Bernstein, D. Leites, Sel. Math. Sov., 1:2 (1981), 143–160 | MR | Zbl