Loop W(a,b) Lie conformal algebra

TL;DR

This paper constructs and analyzes the structure of the loop W(a,b) Lie conformal algebra, including derivations, modules, and extensions, providing new insights into its algebraic properties and representations.

Contribution

It introduces the conformal algebra CLW(a,b) associated with the loop W(a,b) Lie algebra and studies its derivations, modules, and extensions, which is a novel comprehensive analysis.

Findings

Determined the conformal derivations of CLW(a,b).

Classified rank one conformal modules of CLW(a,b).

Analyzed the central extensions and module extensions of CLW(a,b).

Abstract

Fix $a,b\in\C$, let $LW(a,b)$ be the loop $W(a,b)$ Lie algebra over $\C$ with basis $\{L_{\a,i},I_{\b,j} \mid \a,\b,i,j\in\Z\}$ and relations $[L_{\a,i},L_{\b,j}]=(\a-\b)L_{\a+\b,i+j}, [L_{\a,i},I_{\b,j}]=-(a+b\a+\b)I_{\a+\b,i+j},[I_{\a,i},I_{\b,j}]=0$, where $\a,\b,i,j\in\Z$. In this paper, a formal distribution Lie algebra of $LW(a,b)$ is constructed. Then the associated conformal algebra $CLW(a,b)$ is studied, where $CLW(a,b)$ has a $\C[\partial]$-basis $\{L_i,I_j\,|\,i,j\in\Z\}$ with $\lambda$-brackets $[L_i\, {}_\lambda \, L_j]=(\partial+2\lambda) L_{i+j}, [L_i\, {}_\lambda \, I_j]=(\partial+(1-b)\lambda) I_{i+j}$ and $[I_i\, {}_\lambda \, I_j]=0$. In particular, we determine the conformal derivations and rank one conformal modules of this conformal algebra. Finally, we study the central extensions and extensions of conformal modules.