On cohomology of the Lie superalgebra D(2, 1 ; α)

We describe the infinitesimal deformations of the standard embedding of the Lie superalgebra $D(2, 1 ; \alpha)$ into the Poisson superalgebra of pseudodifferential symbols on $S^{1|2}$. We show that for the standard embedding of $D(2, 1 ; \alpha)$ into the Poisson superalgebra of differential operators on $S^{1|2}$, the infinitesimal deformations correspond to formal deformations. For the embedding of $D(2, 1 ; \alpha)$ into the derived contact superconformal algebra ${K}'(4)$, the infinitesimal deformations are formal deformations.