A class of graded conformal algebras which is induced by Heisenberg-Virasoro conformal algebra

TL;DR

This paper classifies a new class of $Z$-graded conformal algebras induced by the Heisenberg-Virasoro algebra, exploring their structure, modules, and derivations, expanding understanding of conformal algebra representations.

Contribution

It introduces and classifies a novel class of $Z$-graded conformal algebras based on the Heisenberg-Virasoro algebra, including module and derivation characterizations.

Findings

Classified $Z$-graded conformal algebras induced by Heisenberg-Virasoro algebra.

Proved all finite irreducible modules are rank-one free modules.

Determined conformal derivations of these graded Lie conformal algebras.

Abstract

In this paper, we obtain a class of $\mathbb{Z}$-graded conformal algebras which is induced by Heisenberg-Virasoro conformal algebra. More precisely, we classify $\mathbb{Z}$-graded conformal algebras $\mathcal{A} = \oplus^\infty_{i=-1}\mathcal{A}_i$ satisfying the following conditions, (C1) $\mathcal{A}_0$ is the Heisenberg-Virasoro conformal algebra; C2) Each $\mathcal{A}_i$ for $i\in\mathbb{Z}_{\ge-1}^*$ is an $\mathcal{A}_0$-module of rank one; (C3) $[{X_{-1}}_\lambda X_i]\neq 0$ for $i\ge 0$, where $X_i$ is any one of $\mathbb{C}[\partial]$-generators of $\mathcal{A}_i$ for $i\in \mathbb{Z}_{\ge -1}$. Further, we prove that all finite nontrivial irreducible modules of these algebras under some special conditions are free of rank one as a $\mathbb{C}[\partial]$-module. The conformal derivations of this class of graded Lie conformal algebras are also determined.