Irreducible modules over the Lie conformal algebra \({\mathfrak {B}}(\alpha ,\beta ,p)\)

In this paper, we introduce a class of infinite Lie conformal algebras \({\mathfrak {B}}(\alpha ,\beta ,p)\) $\mathfrak{B}(\alpha ,\beta ,p)$ , which are the semi-direct sums of Block type Lie conformal algebra \({\mathfrak {B}}(p)\) $\mathfrak{B}(p)$ and its non-trivial conformal modules of \({\mathbb {Z}}\) $\mathbb{Z}$ -graded free intermediate series. The annihilation algebras are a class of infinite-dimensional Lie algebras, which include a lot of interesting subalgebras: Virasoro algebra, Block type Lie algebra, twisted Heisenberg–Virasoro algebra and so on. We give a complete classification of all finite non-trivial irreducible conformal modules of \({\mathfrak {B}}(\alpha ,\beta ,p)\) $\mathfrak{B}(\alpha ,\beta ,p)$ for \(\alpha ,\beta \in {\mathbb {C}}, p\in {\mathbb {C}}^*\) $\alpha ,\beta \in \mathbb{C},p\in {\mathbb{C}}^{\ast }$ . As an application, the classifications of finite irreducible conformal modules over a series of finite Lie conformal algebras \({\mathfrak {b}}(n)\) $\mathfrak{b}(n)$ for \(n\ge 1\) $n\ge 1$ are given.