On the Cohomology of the Lie Superalgebra of Contact Vector Fields on S 1|m |
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Authors: | B Agrebaoui S Mansour |
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Institution: | 1. Département de Mathématiques, Faculté des Sciences de Sfax , Sfax, Tunisie B.Agreba@fss.rnu.tn;3. Département de Mathématiques, Faculté des Sciences de Sfax , Sfax, Tunisie |
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Abstract: | The main result of this article is the explicit calculation of the first cohomology space H 1(𝒦(3), 𝒮Ψ𝒟𝒪(S 1|3)) of the Lie superalgebra 𝒦(3) of contact vector fields on the supercircle S 1|3 with coefficients in the module of superpseudodifferential operators 𝒮Ψ𝒟𝒪(S 1|3). For the supercicles of dimensional 1 | 0, 1 | 1, and 1 | 2, the first cohomology space is computed, respectively, in the following articles: 2
Agrebaoui , B. ,
Ben Fraj , N. ( 2004 ). On the cohomology of the Lie superalgebra of contact vector fields on S 1|1 . Belletin de la Société Royale des Sciences de Liège 72 ( 6 ): 365 – 375 . Google Scholar], 3
Agrebaoui , B. ,
Ben Fraj , N. ,
Omri , S. ( 2006 ). On the cohomology of the Lie superalgebra of contact vector fields on S 1|2 . J. Nonlinear Math. Phys. 13 ( 4 ): 523 – 534 .Taylor &; Francis Online] , Google Scholar], 14
Ovsienko , V. ,
Roger , C. ( 1999 ). Deforming the Lie algebra of vector fields on S 1 inside the Lie algebra of pseudodifferential operators on S 1 . AMS Transl. Ser. 2, (Adv. Math. Sci.) 194 : 211 – 227 . Google Scholar]]. The case m ≥ 4 is still out of reach, but we give a lower bound for the dimension of the cohomology space and exhibit three nontrivial, 1-cocycles. |
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Keywords: | Cohomology Contact vector fields Pseudodifferential operator Supercircle |
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