New Irreducible Modules for Heisenberg and Affine Lie Algebras

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.