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

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_λ L\right]=(\partial+2 λ) (L+I),\ \left[L_λ I\right]=(\partial+λ) I, \ \left[I_λ L\right]=λI,\ \left[I_λ 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.