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.
Categories: math.RA