Extensions of modules over a class of Lie conformal algebras $\mathcal{W}(b)$

TL;DR

This paper classifies the extensions between finite irreducible modules over a specific class of rank 2 Lie conformal algebras, expanding understanding of their module structure.

Contribution

It provides a complete classification of module extensions over the Lie conformal algebra family al{W}(b), a new result in the representation theory of these algebras.

Findings

Classified all extensions between finite irreducible modules over al{W}(b)

Identified conditions for non-trivial extensions

Enhanced understanding of module structures over al{W}(b)

Abstract

Let $\mathcal{W}(b)$ be a class of free Lie conformal algebras of rank $2$ with $\mathbb{C}[\partial]$-basis ${L,H}$ and relations \begin{eqnarray*} [L_\lambda L]=(\partial+2\lambda)L,\ \ [L_\lambda H]=\big(\partial+(1-b)\lambda\big)H, \ \ [H_\lambda H]=0, \end{eqnarray*} where $b$ is a nonzero complex number. In this paper, we classify extensions between two finite irreducible conformal modules over the Lie conformal algebras $\mathcal{W}(b)$.