On modules for double affine Lie algebras

TL;DR

This paper constructs and analyzes various modules for double affine Lie algebras, extending known results from affine Lie algebras, and investigates their properties and relations.

Contribution

It introduces new modules for double affine Lie algebras using multiple triangular decompositions and generalizes key properties from affine Lie algebra theory.

Findings

Construction of imaginary Verma modules and their variants

Relations between different modules are established

Criteria for irreducibility and integrability are provided

Abstract

Imaginary Verma modules, parabolic imaginary Verma modules, and Verma modules at level zero for double affine Lie algebras are constructed using three different triangular decompositions. Their relations are investigated, and several results are generalized from the affine Lie algebras. In particular, imaginary highest weight modules, integrable modules, and irreducibility criterion are also studied.