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

Abstract

In this paper, we introduce a class of infinite Lie conformal algebras \({\mathfrak {B}}(\alpha ,\beta ,p)\), which are the semi-direct sums of Block type Lie conformal algebra \({\mathfrak {B}}(p)\) and its non-trivial conformal modules of \({\mathbb {Z}}\)-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 \({\mathfrak {B}}(\alpha ,\beta ,p)\) for \(\alpha ,\beta \in {\mathbb {C}}, p\in {\mathbb {C}}^*\). As an application, the classifications of finite irreducible conformal modules over a series of finite Lie conformal algebras \({\mathfrak {b}}(n)\) for \(n\ge 1\) are given.