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

In this paper, we classify the Lie conformal superalgebras R=R0̄⊕R1̄, where R1̄ is of rank 1 and R0̄ is the Heisenberg–Virasoro Lie conformal algebra HV, which is a free C[∂]-module generated by L and H satisfying [LλL] = (∂ + 2λ)L, [LλH] = (∂ + λ)H, [HλH] = 0. Based on this, we construct two classes of Lie conformal superalgebras denoted by HVS(α) and 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 HVS(α)0̄=HVS(β,γ,τ)0̄ is HV. Then we study the structure theory of HVS(α) and HVS(β,γ,τ), and completely determine their conformal derivations, conformal biderivations and automorphism groups. Furthermore, we give a classification of the conformal modules of rank (1 + 1) over these two Lie conformal superalgebras.