Loop Heisenberg-Virasoro Lie Conformal algebra

TL;DR

This paper constructs and analyzes the conformal algebra associated with the loop Heisenberg-Virasoro Lie algebra, classifying its derivations and modules, thus advancing the understanding of its algebraic structure.

Contribution

It introduces the conformal algebra $CHV$ from the loop Heisenberg-Virasoro Lie algebra and classifies its derivations and modules, providing new insights into its structure.

Findings

Constructed the conformal algebra $CHV$ from the loop Heisenberg-Virasoro Lie algebra.

Determined the conformal derivations of $CHV$.

Classified rank one conformal modules and $ extbf{Z}$-graded free intermediate series modules.

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 $\{L_i,H_i\,|\,i\in\Z\}$ 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 $\Z$-graded free intermediate series modules over $CHV$ are classified.