On osp(2|2)-relative cohomology of the Lie superalgebra of contact vector fields and deformations

Abstract

Abstract We consider the 1 | 2 -dimensional real superspace R 1 | 2 endowed with its standard contact structure defined by the 1-form α . The conformal Lie superalgebra K ( 2 ) acts on R 1 | 2 as the Lie superalgebra of contact vector fields; it contains the Mobius superalgebra o s p ( 2 | 2 ) . We classify o s p ( 2 | 2 ) -invariant superskew-symmetric binary differential operators from K ( 2 ) ∧ K ( 2 ) to D λ , μ vanishing on o s p ( 2 | 2 ) , where D λ , μ is the superspace of linear differential operators between the superspaces of weighted densities. This result allows us to compute the second differential o s p ( 2 | 2 ) -relative cohomology of K ( 2 ) with coefficients in D λ , μ . We study generic formal o s p ( 2 | 2 ) -trivial deformations of the K ( 2 ) -module structure on the direct sum of the superspaces of weighted densities. This work is the simplest superization of a result by Bouarroudj (2007).