On cohomology of the Lie superalgebra D(2, 1 ; \alpha)

TL;DR

This paper investigates the infinitesimal deformations of the Lie superalgebra D(2, 1 ; lpha) when embedded into certain Poisson and contact superalgebras, revealing their deformation structures.

Contribution

It provides a detailed description of infinitesimal deformations of D(2, 1 ; lpha) embeddings into Poisson superalgebras and the derived contact superconformal algebra.

Findings

Infinitesimal deformations correspond to formal deformations for the standard embedding.

Infinitesimal deformations are also formal for the embedding into the derived contact superconformal algebra.

The work characterizes deformation types of D(2, 1 ; lpha) in specific superalgebra contexts.

Abstract

We describe the infinitesimal deformations of the standard embedding of the Lie superalgebra $D(2, 1 ; \alpha)$ into the Poisson superalgebra of pseudodifferential symbols on $S^{1|2}$. We show that for the standard embedding of $D(2, 1 ; \alpha)$ into the Poisson superalgebra of differential operators on $S^{1|2}$, the infinitesimal deformations correspond to formal deformations. For the embedding of $D(2, 1 ; \alpha)$ into the derived contact superconformal algebra ${K}'(4)$, the infinitesimal deformations are formal deformations.