Loop Heisenberg-Virasoro Lie Conformal algebra

Abstract

Let $HV$ be the loop Heisenberg-Virasoro Lie algebra over $\C$ with basis $\{L_{\a,i},H_{\b,j}\,|\,\a,\,\b,i,j\in\Z\}$ and brackets $[L_{\a,i},L_{\b,j}]=(\a-\b)L_{\a+\b,i+j}, [L_{\a,i},H_{\b,j}]=-\b H_{\a+\b,i+j},[H_{\a,i},H_{\b,j}]=0$. In this paper, a formal distribution Lie algebra of $HV$ is constructed. Then the associated conformal algebra $CHV$ is studied, where $CHV$ has a $\C[\partial]$-basis $\{L_i,H_i|i\in \mathbb{Z}\}$ with $\lambda$-brackets $[L_i{}_{\lambda }{L}_j]=(\partial +2\lambda )L_{i+j},[L_i{}_{\lambda }{H}_j]=(\partial +\lambda )H_{i+j},[H_i{}_{\lambda }{L}_j]=\lambda L_{i+j}$ and $[H_i{}_{\lambda }{H}_j]=0$. In particular, the conformal derivations of $CHV$ are determined. Finally, rank one conformal modules and $\mathbb{Z}$-graded free intermediate series modules over $CHV$ are classified.

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• Infinite rank Schrodinger-Virasoro type Lie conformal algebras