Loop W(a,b) Lie conformal algebra

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.