New Irreducible Modules for Heisenberg and Affine Lie Algebras

TL;DR

This paper constructs new irreducible modules for Heisenberg and affine Lie algebras, expanding the understanding of their representation theory through explicit examples and irreducibility proofs.

Contribution

It introduces new families of irreducible modules over Heisenberg and affine Lie algebras, including explicit constructions and irreducibility results for generalized loop modules.

Findings

Constructed new irreducible modules over Heisenberg Lie algebras.

Proved irreducibility of generalized loop modules over affine Lie algebras.

Established irreducibility of $ ext{phi}$-imaginary Verma modules of nonzero level.

Abstract

We study $\mathbb Z$-graded modules of nonzero level with arbitrary weight multiplicities over Heisenberg Lie algebras and the associated generalized loop modules over affine Kac-Moody Lie algebras. We construct new families of such irreducible modules over Heisenberg Lie algebras. Our main result establishes the irreducibility of the corresponding generalized loop modules providing an explicit construction of many new examples of irreducible modules for affine Lie algebras. In particular, to any function $\phi:\mathbb N\rightarrow \{\pm\}$ we associate a $\phi$-highest weight module over the Heisenberg Lie algebra and a $\phi$-imaginary Verma module over the affine Lie algebra. We show that any $\phi$-imaginary Verma module of nonzero level is irreducible.