First cohomology space of the orthosymplectic Lie superalgebra $\mathfrak{osp}(n|2)$ in the Lie superalgebra of superpseudodifferential operators

Keywords: Cohomology, Orthosymplectic superalgebra, superpseudodifferential operators, Poisson superalgebra

Abstract

UDC 515.12
We investigate the first cohomology space associated with the embedding of the Lie orthosymplectic superalgebra $\mathfrak{osp}(n|2)$ on the $(1,n)$-dimensional superspace $\mathbb{R}^{1|n}$ in the Lie superalgebra $ \mathcal{S}\Psi\mathcal{DO}(n)$ (for $n \geq 4$ ) of superpseudodifferential operators with smooth coefficients.
Following Ovsienko and Roger, we give explicit expressions of the basis cocycles.
This work is the simplest generalization of a result by Basdouri [First space cohomology of the orthosymplectic Lie superalgebra in the Lie superalgebra of superpseudodifferential operators, Algebras and Representation Theory, 16, 35-50 (2013)].

References

B. Agrebaoui, N. Ben Fraj, On the cohomology of Lie superalgebra of contact vector fields on $S^{1/1}$, Bull. Soc. Roy. Sci. Liege, 73 (2004).

B. Agrebaoui, N. Ben Fraj, S. Omri, On the cohomology of Lie superalgebra of contact vector fields on $S^{1|2}$, J. Nonlinear Math. Phys., 13, № 4, 523 – 534 (2006), https://doi.org/10.2991/jnmp.2006.13.4.7

B. Agrebaoui, I. Basdouri, N. Elghomdi, S. Hammami, First space cohomology of the orthosymplectic Lie

superalgebra $frak{osp}(3|2)$ in the Lie superalgebra of superpseudodifferential operators, J. Pseudo-Different. Oper. and Appl., 7, 141 – 155 (2016), https://doi.org/10.1007/s11868-015-0140-x

I. Basdouri, First space cohomology of the orthosymplectic Lie superalgebra in the Lie superalgebra of superpseudodifferential operators, Algebras and Representation Theory, 16, 35 – 50 (2013); https://doi.org/10.1007/s10468-011-9292-4.

M. Ben Ammar, N. Ben Fraj, S. Omri, The binary invariant differential operators on weighted densities on the superspace $Bbb R^{1|n}$ and cohomology, J. Math. Phys., 51, № 4 (2009); https://doi.org/1063/1.3355127.

N. Ben Fraj, S. Omri, Deforming the Lie superalgebra of contact vector fields on $S^{1|1}$, J. Nonlinear Math. Phys., 13, № 1, 19 – 33 (2006), https://doi.org/10.2991/jnmp.2006.13.1.3

N. Ben Fraj, S. Omri, Deformating the Lie superalgebra of contact vector fields on $S^{1|2}$ inside the Lie superalgebra of pseudodifferential operators on $S^{1|2}$, Theore. and Math. Phys., 163, № 2, 618 – 633 (2010).

N. El Gomdi and R. Messaoud, Cohomology of orthosymplectic Lie superalgebra acting on $lambda$ -densities on $R^{1|n}$, Int. J. Geom. Methods Mod. Phys., 14, Issue 01 (2017), https://doi.org/10.1142/S0219887817500165

A. Fialowski, An example of formal deformations of Lie algebras, Proc. NATO, Conf. Deformations Theory of Algebras, Kluwer (1988), p. 3.

A. Fialowski, M. de Montigny, On deformations and contractions of Lie algebras, SIGMA, 2, Article 048 (2006), https://doi.org/10.3842/SIGMA.2006.048

B. L. Feigin, D. B. Fuks, Homology of the Lie algebra of vector fields on the line, Funct. Anal. and Appl., 14, 201 – 212 (1980).

D. B. Fuchs, Cohomology of infinite-dimensional Lie algebras, Plenum Publ., New York (1986).

E. Inonu, E. P. Wigner, On the contraction of groups and their representations, Proc. Nat. Acad. Sci. USA, 39, № 6, 510 – 524 (1953), https://doi.org/10.1073/pnas.39.6.510

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, 211 – 226 (1999), https://doi.org/10.1090/trans2/194/09

V. Ovsienko, C. Roger, Deforming the Lie algebra of vector fields on $S^1$ inside the Poisson algebra on $dot T{}^ast S^1$, Comm. Math. Phys., 198, 97 – 110 (1998), https://doi.org/10.1007/s002200050473

I. E. Segal, A class of operator algebras which are determined by groups, Duke Math. J., 18, № 1, 221 – 265 (1951).

E. J. Saletan, Contraction of Lie groups, J. Math. Phys., 2, 1 – 21 (1961), https://doi.org/10.1063/1.1724208

Published
07.07.2022
How to Cite
Boujelben, M. “First Cohomology Space of the Orthosymplectic Lie Superalgebra $\mathfrak{osp}(n|2)$ in the Lie Superalgebra of Superpseudodifferential Operators”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, no. 6, July 2022, pp. 761 -71, doi:10.37863/umzh.v74i6.6052.
Section
Research articles