On matrix realizations of the contact superconformal algebra $\hat{K}'(4)$ and the exceptional N = 6 superconformal algebra

TL;DR

This paper provides matrix realizations of the superalgebras $ ilde{K}'(4)$ and the exceptional N=6 superconformal algebra, linking their representations to embeddings into pseudodifferential symbol superalgebras.

Contribution

It introduces matrix realizations of these superalgebras over a Weyl algebra, clarifying their structure and representations in specific superspaces.

Findings

Realizations of $ ilde{K}'(4)$ and N=6 superconformal algebra as matrices over Weyl algebra.

Explicit construction of representations in superspaces $V^{ ext{mu}}$ for $ ext{mu} = 0$.

Connection of these realizations to embeddings into pseudodifferential symbol superalgebras.

Abstract

The superalgebra $\hat{K}'(4)$ and the exceptional N = 6 superconformal algebra have ``small'' irreducible representations in the superspaces $V^{\mu} = t^{\mu}\C[t, t^{-1}]\otimes\Lambda(N)$, where N = 2 and 3, respectively. For ${\mu \in \C\backslash \Z}$ they are associated to the embeddings of these superalgebras into the Lie superalgebras of pseudodifferential symbols on the supercircle S^{1|N}. In this work we describe $\hat{K}'(4)$ and the exceptional N = 6 superconformal algebra in terms of matrices over a Weyl algebra. Correspondingly, we obtain realizations of their representations in $V^{\mu}$ for $\mu = 0$.