Geodesic
Deforming the Lie superalgebra of contact vector fields on $S^{1|2}$ inside the Lie superalgebra of pseudodifferential operators on $S^{1|2}$
Teoretičeskaâ i matematičeskaâ fizika, Tome 163 (2010) no. 2, pp. 258-276 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We classify deformations of the standard embedding of the Lie superalgebra $\mathcal K(2)$ of contact vector fields on the $(1,2)$-dimensional supercircle into the Lie superalgebra $S\Psi D(S^{1|2})$ of pseudodifferential operators on the supercircle $S^{1|2}$. The proposed approach leads to the deformations of the central charge induced on $\mathcal K(2)$ by the canonical central extension of $S\Psi D(S^{1|2})$.
Keywords: cohomology, Lie superalgebra, Lie superalgebra deformation. 
@article{TMF_2010_163_2_a2,
     author = {N. Ben Fraj and S. Omri},
     title = {Deforming {the~Lie} superalgebra of contact vector fields on $S^{1|2}$ inside {the~Lie} superalgebra of pseudodifferential operators on $S^{1|2}$},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {258--276},
     year = {2010},
     volume = {163},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2010_163_2_a2/}
}
TY  - JOUR
AU  - N. Ben Fraj
AU  - S. Omri
TI  - Deforming the Lie superalgebra of contact vector fields on $S^{1|2}$ inside the Lie superalgebra of pseudodifferential operators on $S^{1|2}$
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2010
SP  - 258
EP  - 276
VL  - 163
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TMF_2010_163_2_a2/
LA  - ru
ID  - TMF_2010_163_2_a2
ER  - 
%0 Journal Article
%A N. Ben Fraj
%A S. Omri
%T Deforming the Lie superalgebra of contact vector fields on $S^{1|2}$ inside the Lie superalgebra of pseudodifferential operators on $S^{1|2}$
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2010
%P 258-276
%V 163
%N 2
%U http://geodesic.mathdoc.fr/item/TMF_2010_163_2_a2/
%G ru
%F TMF_2010_163_2_a2
N. Ben Fraj; S. Omri. Deforming the Lie superalgebra of contact vector fields on $S^{1|2}$ inside the Lie superalgebra of pseudodifferential operators on $S^{1|2}$. Teoretičeskaâ i matematičeskaâ fizika, Tome 163 (2010) no. 2, pp. 258-276. http://geodesic.mathdoc.fr/item/TMF_2010_163_2_a2/

[1] P. B. Cohen, Yu. I. Manin, D. Zagier, “Automorphic pseudodifferential operators”, Algebraic Aspects of Integrable Systems, Progr. Nonlinear Differential Equations Appl., 26, eds. A. S. Fokas, I. M. Gelfand, Brikhäuser, Boston, MA, 1997, 17–47 | MR

[2] D. R. Lebedev, Yu. I. Manin, Phys. Lett. A, 74:3–4 (1979), 154–156 | DOI | MR

[3] D. R. Lebedev, Yu. I. Manin, Funkts. analiz i ego pril., 13:4 (1979), 40–46 | DOI | MR | Zbl

[4] V. Ovsienko, C. Roger, “Deforming the Lie algebra of vector fields on $S^1$ inside the Lie algebra of pseudodifferential symbols on $S^1$”, Differential Topology, Infinite-Dimensional Lie Algebras, and Applications, Amer. Math. Soc. Transl. Ser. 2, 194, eds. A. Astashkevich S. Tabachnikov, AMS, Providence, RI, 1999, 211–227 | MR | Zbl

[5] V. Ovsienko, C. Roger, Comm. Math. Phys., 198:1 (1998), 97–110 | DOI | MR | Zbl

[6] M. Adler, Invent. Math., 50:3 (1979), 219–248 | DOI | MR | Zbl

[7] P. Grozman, D. Leites, I. Shchepochkina, Acta Math. Vietnam., 26:1 (2001), 27–63 ; arXiv: hep-th/9702120 | MR | Zbl

[8] N. Ben Fraj, S. Omri, J. Nonlinear Math. Phys., 13:1 (2006), 19–33 | DOI | MR | Zbl

[9] D. B. Fuchs, D. A. Leites, C. R. Acad. Bulgare Sci., 37:12 (1985), 1595–1596 | MR | Zbl

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

[11] A. Dzhumadil'daev, Z. Phys. C, 72:3 (1996), 509–517 | DOI | MR | Zbl

[12] E. Poletaeva, J. Math. Phys., 42:1 (2001), 526–540 | DOI | MR | Zbl

[13] A. O. Radul, Phys. Lett. B, 265:1–2 (1991), 86–91 | DOI | MR

[14] Yu. I. Manin, A. O. Radul, Comm. Math. Phys., 98:1 (1985), 65–77 | DOI | MR | Zbl

[15] Quantum Fields and Strings: a Course for Mathematicians, V. 1, 2. Material from the Special Year on Quantum Field Theory (Institute for Advanced Study, Princeton, NJ, 1996–1997), eds. P. Deligne, P. Etingof, D. Freed, L. Jeffrey, D. Kazhdan, J. Morgan, D. Morrison, E. Witten, AMS, Providence, RI, 1999 | MR | Zbl

[16] B. M. Zupnik, D. G. Pak, TMF, 77:1 (1988), 97–106 | DOI | MR

[17] A. Fialowski, D. B. Fuchs, J. Funct. Anal., 161:1 (1999), 76–110 | DOI | MR | Zbl

[18] A. Nijenhuis, R. W. Richardson Jr., Bull. Amer. Math. Soc., 73 (1967), 175–179 | DOI | MR | Zbl

[19] B. Agrebaoui, F. Ammar, P. Lecomte, V. Ovsienko, Internat. Math. Res. Not., 2002:16 (2002), 847–869 | DOI | MR | Zbl

[20] B. Agrebaoui, M. Ben Ammar, N. Ben Fraj, V. Ovsienko, J. Nonlinear Math. Phys., 10:2 (2003), 148–156 | DOI | MR | Zbl

[21] M. Ben Ammar, M. Boujelbene, SIGMA, 4 (2008), 065 | DOI | MR | Zbl

[22] B. Agrebaoui, N. Ben Fraj, S. Omri, J. Nonlinear Math. Phys., 13:4 (2006), 523–534 ; arXiv: math-ph/0603057 | DOI | MR | Zbl