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

We obtain a realization of the Lie superalgebra D(2,1;α)D(2,1;α) in differential operators on the supercircle S1|2S1|2 and in 4×4 matrices over a Weyl algebra. A contraction of D(2,1;α)D(2,1;α) is isomorphic to the universal central extension pslˆ(2|2) of psl(2|2)psl(2|2). We realize it in 4×4 matrices over the associative algebra of pseudodifferential operators on S1S1. Correspondingly, there exists a three-parameter family of irreducible representations of pslˆ(2|2) in a (2|2)(2|2)-dimensional complex superspace.