The second cohomology spaces $\mathcal{K}(2)$ with coefficents in the superspace of weighted densities

Keywords: Cohomologie, Lie superalgebra, weighted densities


UDC 515.1

Over the $(1,2)$-dimensional supercircle, we investigate the second cohomology space associated the lie superalgebra $\mathcal{K}(2)$ of vector fields on the supercircle $S^{1|2}$ with coefficients in the space of weighted densities.
We explicitly give 2-cocycle spanning these cohomology spaces.



B. Agrebaoui, I. Basdouri, M. Boujelben, The second cohomology spaces of $K(1)$ with coefficient in the superspace of weighted densities and deformation of the superspace of symbols on $S^{1|1}$, Submetted, hal-01699198, v1.

I. Basdouri, On $osp(1|2)$-relative cohomology on $S^{1|1}$, Commun. Algebra, 42, № 4, 1698 – 1710 (2014), DOI: 10.1080/00927872.2012.748782

I. Basdouri, M. Ben Ammar, Cohomology of $osp (1| 2)$ acting on linear differential operators on the supercercle $S^{1|1}$, Lett. Math. Phys., 81, № 3, 239 – 251 (2007), DOI: 10.1007/s11005-007-0181-z

V. Ovsienko, C. Roger, Extension of Virasoro group and Virasoro algebra by modules of tensor densities on $S^1$, Funct. Anal. and Appl., 30, № 4, 290 – 291 (1996), DOI: 10.1007/BF02509628

V. Ovsienko, P. Marcel, Extension of the Virasoro and Neveu – Schwartz algebras and generalized Sturm – Liouville operators, Lett. Math. Phys., 40, № 1, 31 – 39 (1997), DOI: 10.1007/s11005-007-0181-z

How to Cite
Basdouri, O., A. Braghtha, and S. Hammami. “The Second Cohomology Spaces $\mathcal{K}(2)$ With coefficents in the Superspace of Weighted Densities”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, no. 10, Oct. 2020, pp. 1323-34, doi:10.37863/umzh.v72i10.6030.
Research articles