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

Abstract

We obtain a realization of the Lie superalgebra $D(2,1;\alpha )$ in differential operators on the supercircle $S^{1|2}$ and in $4\times 4$ matrices over a Weyl algebra. A contraction of $D(2,1;\alpha )$ is isomorphic to the universal central extension $\hat{\p\sł}(2|2)$ of $\p\sł(2|2)$. We realize it in $4\times 4$ matrices over the associative algebra of pseudodifferential operators on $S^1$. Correspondingly, there exists a three-parameter family of irreducible representations of $\hat{\p\sł}(2|2)$ in a $(2|2)$--dimensional complex superspace.

15 pages, to be published in Journal of Geometry and Physics 60 (2010), 1656-1664

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