Finite irreducible conformal modules over the extended Block type Lie conformal algebra $\mathfrak{B}(\alpha,\beta,p)$

TL;DR

This paper classifies all finite non-trivial irreducible conformal modules over a new class of infinite Lie conformal algebras $rak{B}( ext{parameters})$, extending understanding of their representation theory and related algebraic structures.

Contribution

It provides a complete classification of finite irreducible conformal modules over the extended Block type Lie conformal algebra $rak{B}( ext{parameters})$, including applications to finite Lie conformal algebras.

Findings

Complete classification of finite irreducible conformal modules over $rak{B}( ext{parameters})$

Identification of subalgebras such as Virasoro and Block types within annihilation algebras

Applications to finite Lie conformal algebras $rak{b}(n)$ for $n extgreater 1$

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 $\Z$-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 $\alpha,\beta\in\C, p\in\C^*$. As an application, the classifications of finite irreducible conformal modules over a series of finite Lie conformal algebras $\mathfrak{b}(n)$ for $n\geq1$ are given.