# Second Cohomology Space of the Lie Superalgebra of contact vector fields $\mathcal{K}(2)$ with coefficients in the superspace of weighted densities on $S^{1|2}$

## TL;DR

This paper computes the second cohomology space of the Lie superalgebra of contact vector fields on a (1,2)-dimensional superspace, providing explicit cocycles that span this space, advancing understanding of its structure.

## Contribution

It explicitly determines the second cohomology space of $rak{K}(2)$ with coefficients in weighted densities, including explicit cocycles, which was previously unknown.

## Key findings

- Explicit cocycles spanning the second cohomology space are provided.
- The structure of the second cohomology space for $rak{K}(2)$ is characterized.
- The results contribute to the understanding of deformations and extensions of the superalgebra.

## Abstract

We investigate the second cohomology space of the Lie superalgebra $\mathcal{K}(2)$ with coefficients in the superspace of weighted densities on the (1,2)-dimentional real superspace. We explicitly give cocycles spanning this cohomology space.