ABSTRACT
Let osp ( 2 | 2 ) be the orthosymplectic Lie superalgebra and osp ( 1 | 2 ) a Lie subalgebra of osp ( 2 | 2 ) . In our paper, we describe the cup-product H 1 ⨠H 1 , where H 1 : = H 1 ( osp ( 2 | 2 ) , osp ( 1 | 2 ) ; D λ , µ 2 ) is the first differential osp ( 1 | 2 ) -relative cohomology of osp ( 2 | 2 ) with coefficients in D λ , µ 2 and D λ , µ 2 : = Ho m diff ( F λ 2 , F µ 2 ) is the space of linear differential operators acting on weighted densities. This result allows us to classify the osp ( 1 | 2 ) -trivial deformations of the osp ( 2 | 2 ) -module structure on the spaces of symbols S d 2 . More precisely, we compute the necessary and sufficient integrability conditions of a given infinitesimal deformation of this action. Furthermore, we prove that any formal osp ( 1 | 2 ) -trivial deformations of osp ( 2 | 2 ) -modules of symbols is equivalent to its infinitisemal part. This work is the simplest generalization of a result by Laraiedh [17].
