Indecomposable modules of the intermediate series over W(a,b) algebras

TL;DR

This paper classifies indecomposable modules of the intermediate series over W(a,b) algebras and characterizes irreducible Harish-Chandra modules, revealing their structure and conditions for simplicity.

Contribution

It provides a complete classification of indecomposable modules over W(a,b) and characterizes irreducible Harish-Chandra modules, including their relation to Vir-modules.

Findings

Indecomposable modules of the intermediate series are classified.

Irreducible Harish-Chandra modules are either highest/lowest weight or uniformly bounded.

For irrational a, irreducible weight modules are Vir-modules with trivial W_k actions.

Abstract

For any complex parameters a,b, the W(a,b) algebra is the Lie algebra with basis {L_i,W_i|i\in Z}, and relations [L_i,L_j]=(j-i)L_{i+j}, [L_i,W_j]=(a+j+bi)W_{i+j},[W_i,W_j]=0. In this paper, indecomposable modules of the intermediate series over W(a,b) are classified. It is also proved that an irreducible Harish-Chandra W(a,b)-module is either a highest/lowest weight module or a uniformly bounded module. Furthermore, if a\notin Q, an irreducible weight W(a,b)-module is simply a Vir-module with trivial actions of W_k.