Two classes of Lie conformal superalgebras related to the Heisenberg--Virasoro Lie conformal algebra

TL;DR

This paper introduces two new classes of Lie conformal superalgebras related to the Heisenberg--Virasoro algebra, and fully characterizes their derivations, automorphisms, and modules.

Contribution

It constructs two novel classes of Lie conformal superalgebras and determines their structural properties and module classifications.

Findings

Constructed two classes of Lie conformal superalgebras, alpha and beta,gamma,tau.

Determined conformal derivations, biderivations, automorphism groups, and modules of rank (1+1).

Established the relation of these superalgebras to the Heisenberg--Virasoro algebra.

Abstract

In this paper, firstly we construct two classes of Lie conformal superalgebras denoted by $\mathcal{HVS}(\alpha)$ and $\mathcal{HVS}(\beta,\gamma,\tau)$, respectively, where $\alpha$ is an nonzero complex number and $\beta,\gamma,\tau$ are complex numbers. They are both of rank $(2+1)$ and the even part $\mathcal{HVS}(\alpha)_{\bar{0}}=\mathcal{HVS}(\beta,\gamma,\tau)_{\bar{0}}$ is the Heisenberg--Virasoro Lie conformal algebra, which is a free $\mathbb{C}[\partial]$-module generated by $L$ and $H$ satisfying $[L_\lambda L]=(\partial+2\lambda)L,\ [L_\lambda H]=(\partial+\lambda)H,\ [H_\lambda H]=0$. Then we completely determine the conformal derivations, conformal biderivations, automorphism groups and conformal modules of rank $(1+1)$ for these two Lie conformal superalgebras.