Block type Lie algebras and their representations

TL;DR

This paper introduces a new class of infinite-dimensional Block type Lie algebras, studies their representation theory, and classifies their irreducible modules, extending previous research in the field.

Contribution

It generalizes existing Block type Lie algebras and provides a classification of their irreducible quasifinite modules, including highest, lowest, and bounded modules.

Findings

Classified all quasifinite irreducible modules of $\

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Abstract

Block type Lie algebras have been studied by many authors in the latest twenty years. In this paper, we will study a class of more general Block type Lie algebra $\mathcal{B}(p,q)$, which is a class of infinite-dimensional Lie algebra by using the generalized Balinskii-Novikov's construction method to Witt type Novikov algebra. We study the representation theory for $\mathcal{B}(p,q)$. We classify quasifinite irreducible highest weight $\mathcal{B}(p,q)$-module. We also prove that any quasifinite irreducible module of Block type Lie algebras $\mathcal{B}(p,q)$ is either a highest or lowest weight module, or else a uniformly bounded module. This paper can be considered as a generalization of the related literatures.