Embedding of the Lie superalgebra $D(2, 1 ; \alpha)$ into the Lie superalgebra of pseudodifferential symbols on $S^{1|2}$

TL;DR

This paper constructs an explicit embedding of the exceptional Lie superalgebras $D(2, 1 ; \alpha)$ into the algebra of pseudodifferential symbols on a supercircle, revealing new structural relationships in superconformal algebra representations.

Contribution

It introduces a novel embedding of $D(2, 1 ; \alpha)$ into pseudodifferential symbols and a central extension of the contact superconformal algebra, expanding understanding of superalgebra representations.

Findings

Embedding of $D(2, 1 ; \alpha)$ into pseudodifferential symbols on $S^{1|2}$

Realization of $D(2, 1 ; \alpha)$ in a matrix algebra over a Weyl algebra

Connection to a central extension of the superconformal algebra $K'(4)$

Abstract

We obtain an embedding of a one-parameter family of exceptional simple Lie superalgebras $D(2, 1 ; \alpha)$ into the Lie superalgebra of pseudodifferential symbols on the supercircle $S^{1|2}$. Correspondingly, there is an embedding of $D(2, 1 ; \alpha)$ into a nontrivial central extension of the derived contact superconformal algebra $K'(4)$ realized in terms of $4\times 4$ matrices over a Weyl algebra.