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TMF, 2001, Volume 126, Number 3, Pages 339–369 (Mi tmf435)  

This article is cited in 24 scientific papers (total in 24 papers)

How to Quantize the Antibracket

D. A. Leitesa, I. M. Shchepochkinab

a Stockholm University
b Independent University of Moscow

Abstract: 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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English version:
Theoretical and Mathematical Physics, 2001, 126:3, 281–306

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Received: 08.04.2000
Revised: 02.10.2000

Citation: D. A. Leites, I. M. Shchepochkina, “How to Quantize the Antibracket”, TMF, 126:3 (2001), 339–369; Theoret. and Math. Phys., 126:3 (2001), 281–306

Citation in format AMSBIB
\by D.~A.~Leites, I.~M.~Shchepochkina
\paper How to Quantize the Antibracket
\jour TMF
\yr 2001
\vol 126
\issue 3
\pages 339--369
\jour Theoret. and Math. Phys.
\yr 2001
\vol 126
\issue 3
\pages 281--306

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    This publication is cited in the following articles:
    1. Arik, M, “The anticommutator spin algebra, its representations and quantum group invariance”, International Journal of Modern Physics A, 18:27 (2003), 5039  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    2. Groman P., Leites D., Shchepochkina I., “Defining relations for the exceptional Lie superalgebras of vector fields”, Orbit Method in Geometry and Physics - in Honor of A.A. Kirillov, Progress in Mathematics, 213, 2003, 101–146  mathscinet  isi
    3. J. Math. Sci. (N. Y.), 133:4 (2006), 1464–1476  mathnet  crossref  mathscinet  zmath
    4. Grozman, P, “Lie superalgebra structures in H-center dot (g;g)”, Czechoslovak Journal of Physics, 54:11 (2004), 1313  crossref  mathscinet  adsnasa  isi  scopus  scopus
    5. S. E. Konstein, A. G. Smirnov, I. V. Tyutin, “Cohomologies of the Poisson superalgebra”, Theoret. and Math. Phys., 143:2 (2005), 625–650  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    6. S. E. Konstein, I. V. Tyutin, “Cohomology of the Poisson Superalgebra on Spaces of Superdimension $(2,n_-)$”, Theoret. and Math. Phys., 145:3 (2005), 1619–1645  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    7. Grozman, P, “Structures of G(2) type and nonintegrable distributions in characteristic p”, Letters in Mathematical Physics, 74:3 (2005), 229  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    8. S. E. Konstein, A. G. Smirnov, I. V. Tyutin, “General form of the deformation of the Poisson superbracket”, Theoret. and Math. Phys., 148:2 (2006), 1011–1024  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    9. S. E. Konstein, I. V. Tyutin, “Deformations of the nondegenerate constant Poisson bracket and antibracket on superspaces of an arbitrary superdimension”, Theoret. and Math. Phys., 155:1 (2008), 598–605  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    10. S. E. Konstein, I. V. Tyutin, “General form of the deformation of the Poisson superbracket on a $(2,n)$-dimensional superspace”, Theoret. and Math. Phys., 155:2 (2008), 734–753  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    11. Konstein, SE, “Deformations and central extensions of the antibracket superalgebra”, Journal of Mathematical Physics, 49:7 (2008), 072103  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    12. S. E. Konstein, A. G. Smirnov, I. V. Tyutin, “Hochschild cohomologies and deformations of the pointwise superproduct”, Theoret. and Math. Phys., 158:3 (2009), 271–292  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    13. Sofiane Bouarroudj, Pavel Grozman, Dimitry Leites, “Classification of Finite Dimensional Modular Lie Superalgebras with Indecomposable Cartan Matrix”, SIGMA, 5 (2009), 060, 63 pp.  mathnet  crossref  mathscinet
    14. Popowicz, Z, “Odd Hamiltonian structure for supersymmetric Sawada-Kotera equation”, Physics Letters A, 373:37 (2009), 3315  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    15. Lebedev A., “Analogs of the Orthogonal, Hamiltonian, Poisson, and Contact Lie Superalgebras in Characteristic 2”, J Nonlinear Math Phys, 17, Suppl. 1 (2010), 217–251  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    16. Iyer U.N., Leites D., Messaoudene M., Shchepochkina I., “Examples of Simple Vectorial Lie Algebras in Characteristic 2”, J Nonlinear Math Phys, 17, Suppl. 1 (2010), 311–374  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    17. Batalin I.A., Bering K., “Path integral formulation with deformed antibracket”, Phys Lett B, 694:2 (2010), 158–166  crossref  mathscinet  adsnasa  isi  elib  scopus  scopus
    18. Popowicz Z., “Does the supersymmetric integrability imply the integrability of Bosonic sector”, Nonlinear and Modern Mathematical Physics, AIP Conference Proceedings, 1212, 2010, 50–57  crossref  mathscinet  zmath  isi  scopus  scopus
    19. S. Bouarroudj, A. V. Lebedev, F. Vagemann, “Deformations of the Lie Algebra $\mathfrak{o}(5)$ in Characteristics $3$ and $2$”, Math. Notes, 89:6 (2011), 777–791  mathnet  crossref  crossref  mathscinet  isi
    20. S. E. Konstein, I. V. Tyutin, “Deformations of the antibracket with Grassmann-valued deformation parameters”, Theoret. and Math. Phys., 183:1 (2015), 501–515  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    21. Batalin I.A., Lavrov P.M., “Extended SIGMA-Model in Nontrivially Deformed Field-Antifield Formalism”, Mod. Phys. Lett. A, 30:29 (2015), 1550141  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    22. Bouarroudj S., Leites D., “Invariant Differential Operators in Positive Characteristic”, J. Algebra, 499 (2018), 281–297  crossref  mathscinet  zmath  isi  scopus
    23. Bouarroudj S., Krutov A., Leites D., Shchepochkina I., “Non-Degenerate Invariant (Super)Symmetric Bilinear Forms on Simple Lie (Super)Algebras”, Algebr. Represent. Theory, 21:5 (2018), 897–941  crossref  mathscinet  zmath  isi  scopus
    24. D. A. Leites, “Two problems in the theory of differential equations”, Theoret. and Math. Phys., 198:2 (2019), 271–283  mathnet  crossref  crossref  adsnasa  isi  elib
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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