Journal of Nonlinear Mathematical Physics

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Volume 16, Issue 4, December 2009, Pages 373 - 409

Cohomology of the Lie Superalgebra of Contact Vector Fields on đť•‚1|1 and Deformations of the Superspace of Symbols

Authors
Imed Basdouri*, ‡, Mabrouk Ben Ammar*, §, Nizar Ben Fraj†, ¶, Maha Boujelbene*, ‖, Kaouthar Kamoun*, **
*Département de Mathématiques, Faculté des Sciences de Sfax, Route de Soukra 3018 Sfax BP 802, Tunisie
†Institut Supérieur de Sciences Appliquées et Technologie, Sousse, Tunisie
Received 10 December 2006, Revised 16 April 2009, Accepted 24 April 2009, Available Online 7 January 2021.
DOI
10.1142/S1402925109000431How to use a DOI?
Keywords
Superconformal algebra; cohomology; deformations; differential operators; orthosymplectic superalgebra; contact geometry; tensor densities
Abstract

Following Feigin and Fuchs, we compute the first cohomology of the Lie superalgebra 𝒦(1) of contact vector fields on the (1, 1)-dimensional real or complex superspace with coefficients in the superspace of linear differential operators acting on the superspaces of weighted densities. We also compute the same, but osp(1|2)-relative, cohomology. We explicitly give 1-cocycles spanning these cohomology. We classify generic formal osp(1|2)-trivial deformations of the 𝒦(1)-module structure on the superspaces of symbols of differential operators. We prove that any generic formal osp(1|2)-trivial deformation of this 𝒦(1)-module is equivalent to a polynomial one of degree ≤ 4. This work is the simplest superization of a result by Bouarroudj [On sl(2)-relative cohomology of the Lie algebra of vector fields and differential operators, J. Nonlinear Math. Phys. No. 1 (2007) 112–127]. Further superizations correspond to osp(N|2)-relative cohomology of the Lie superalgebras of contact vector fields on 1|N-dimensional superspace.

Copyright
© 2009 The Authors. Published by Atlantis Press and Taylor & Francis
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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Next Article In Issue>
Journal
Journal of Nonlinear Mathematical Physics
Volume-Issue
16 - 4
Pages
373 - 409
Publication Date
2021/01/07
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
10.1142/S1402925109000431How to use a DOI?
Copyright
© 2009 The Authors. Published by Atlantis Press and Taylor & Francis
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

Cite this article

risenwbib
TY  - JOUR
AU  - Imed Basdouri
AU  - Mabrouk Ben Ammar
AU  - Nizar Ben Fraj
AU  - Maha Boujelbene
AU  - Kaouthar Kamoun
PY  - 2021
DA  - 2021/01/07
TI  - Cohomology of the Lie Superalgebra of Contact Vector Fields on đť•‚Âą|Âą and Deformations of the Superspace of Symbols
JO  - Journal of Nonlinear Mathematical Physics
SP  - 373
EP  - 409
VL  - 16
IS  - 4
SN  - 1776-0852
UR  - https://doi.org/10.1142/S1402925109000431
DO  - 10.1142/S1402925109000431
ID  - Basdouri2021
ER  -