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 $V^μ = t^μ\C[t, t^{-1}]\otimesΛ(N)$, where N = 2 and 3, respectively. For ${μ\in \C\backslash \Z}$ 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 $V^μ$ for $μ= 0$.