Conformal modules and their extensions of a Lie conformal algebra related to a 2-dimensional Novikov algebra

TL;DR

This paper classifies all finite irreducible conformal modules over a specific rank 2 Lie conformal algebra related to a 2-dimensional Novikov algebra and determines the extensions between them.

Contribution

It provides a complete classification of finite irreducible modules and their extensions for a particular Lie conformal algebra associated with a Novikov algebra.

Findings

Classified all finite irreducible conformal modules over the algebra.

Determined all possible extensions between these modules.

Abstract

Let $\mathcal{R}$ be a free Lie conformal algebra of rank $2$ with $\mathbb{C}[\partial]$-basis $\{L,I\}$ and relations \begin{eqnarray*} \left[L_{\lambda} L\right]=(\partial+2 \lambda) (L+I),\ \left[L_{\lambda} I\right]=(\partial+\lambda) I, \ \left[I_{\lambda} L\right]=\lambda I,\ \left[I_{\lambda} I\right]=0. \end{eqnarray*} In this paper, we first classify all finite nontrivial irreducible conformal modules over $\mathcal{R}$. Then we determine extensions between two finite irreducible conformal $\mathcal{R}$-modules.