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

We describe the infinitesimal deformations of the standard embedding of the Lie superalgebra D(2,1;α)D(2,1;α) into the Poisson superalgebra of pseudodifferential symbols on S1∣2S1∣2. We show that for the standard embedding of D(2,1;α)D(2,1;α) into the Poisson superalgebra of differential operators on S1|2S1|2, the infinitesimal deformations correspond to formal deformations. For the embedding of D(2,1;α)D(2,1;α) into the derived contact superconformal algebra K′(4)K′(4), the infinitesimal deformations are formal deformations.