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

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 $\mu = 0$.