Structures of $W(2,2)$ Lie conformal algebra

TL;DR

This paper thoroughly investigates the structure, derivations, extensions, modules, and cohomology of the $W(2,2)$ Lie conformal algebra, providing a comprehensive understanding of its algebraic properties.

Contribution

It offers the first detailed analysis of conformal derivations, central extensions, modules, and cohomology specifically for the $W(2,2)$ Lie conformal algebra.

Findings

Determined conformal derivations and central extensions.

Classified conformal modules for the algebra.

Computed the cohomology groups with trivial coefficients.

Abstract

The purpose of this paper is to study $W(2,2)$ Lie conformal algebra, which has a free $\mathbb{C}[\partial]$-basis $\{L, M\}$ such that $[L_\lambda L]=(\partial+2\lambda)L$, $[L_\lambda M]=(\partial+2\lambda)M$, $[M_\lambda M]=0$. In this paper, we study conformal derivations, central extensions and conformal modules for this Lie conformal algebra. Also, we compute the cohomology of this Lie conformal algebra with coefficients in its modules. In particular, we determine its cohomology with trivial coefficients both for the basic and reduced complexes.