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

Abstract

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