Finite irreducible conformal modules over the extended Block type 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 $α,β\in\C, p\in\C^*$. 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.