On the matrix realization of the Lie superalgebra of contact projective vector fields $\mathfrak{spo}(2l+2|n)$

Abstract

In this paper, we show that the Lie superalgebra $\mathfrak{spo}(2l+2|n)$ is into the intersection of Lie superalgebra of contact vector fields $\mathcal{K}(2l+1|n)$ and the Lie superalgebra of projective vector fields $\mathfrak{pgl}(2l+2|n)$. We use mainly the embedding used by P. Mathonet and F. Radoux in "\textit{ Projectively equivariant quantizations over superspace ${\mathbb{R}}^{p|q}$. Lett. Math. Phys, 98: 311-331, 2011}". Explicitly, we use the embedding of a Lie superalgebra constituted of matrices belonging to $\gl(2l+2|n)$ into $\mathrm{Vect}({\mathbb{R}}^{2l+1|n})$. We generalize thus in superdimension $2l+1-n$, the matrix realization described in \cite{MelNibRad13} on $S^{1|2}$. We mention that the intersection $\spo(2l+2|n)=\pgl(2l+2|n)\cap\cK(2l+1|n)$ that we prove here, in super case, has been prooved on ${\mathbb{R}}^{2l+2}$ in even case in \cite{CoOv12}.