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

TL;DR

This paper constructs explicit matrix and differential operator realizations of the Lie superalgebra D(2, 1 ; lpha), explores its contraction to a central extension of p(2|2), and discusses associated representations.

Contribution

It provides new realizations of D(2, 1 ; lpha) and its contraction in matrix and differential operator forms, along with a family of irreducible representations.

Findings

Realization of D(2, 1 ; lpha) on supercircle using differential operators.

Matrix realization of the contraction as a central extension of p(2|2).

Existence of a three-parameter family of irreducible representations.

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 $\hat{\p\s\l}(2|2)$ of $\p\s\l(2|2)$. 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 $\hat{\p\s\l}(2|2)$ in a $(2|2)$--dimensional complex superspace.