Irreducible representations of untwisted affine Kac-Moody algebras

Abstract

In this paper we construct a class of new irreducible modules over untwisted affine Kac-Moody algebras $\tilde{\mathfrak{g}}$, generalizing and including both highest weight modules and Whittaker modules. These modules allow us to obtain a complete classification of irreducible $\tilde{\mathfrak{g}}$-modules on which the action of each root vector in ${\tilde{\mathfrak{n}}}_{+}$ is locally finite, where ${\tilde{\mathfrak{n}}}_{+}$ is the locally nilpotent subalgebra (or positive part) of $\tilde{\mathfrak{g}}$. The necessary and sufficient conditions for two such irreducible $\tilde{\mathfrak{g}}$-modules to be isomorphic are also determined. In the second part of the paper, we use the "shifting technique" to obtain a necessary and sufficient condition for the tensor product of irreducible integrable loop $\tilde{\mathfrak{g}}$-modules and irreducible integrable highest weight $\tilde{\mathfrak{g}}$-modules to be simple. This tensor product problem was originally studied by Chari and Pressley 28 years ago.

36 pages

References in corpus (2)

• Characterization of simple highest weight modules
• Classification of simple weight modules over affine Lie algebras

Cited by in corpus (5)

• Whittaker modules for the affine Lie algebra $A_1^{(1)}$
• Module structure on $U(\mathfrak{h})$ for Kac-Moody algebras
• $U(\fh)$-free modules over the Block algebra $\BB(q)$
• Tensor product weight modules for the mirror-twisted Heisenberg-Virasoro algebra
• Quasi-Whittaker modules for the Schrödinger algebra