Complete reducibility of the level modules of the rank 3 Kac–Moody algebra

Let ${\mathcal F}ib(m)$ denote the level-$m$ module for the rank 2 hyperbolic subalgebra ${\mathcal F}ib$, and let $|m|\leq 2$ describe the exceptional levels. A trivial module is the one-dimensional module on which ${\mathcal F}ib$ acts trivially; the non-standard modules occur on levels with $|m|\leq 2$, while standard highest- and lowest-weight modules are the remaining modules.

Complete reducibility conjecture. For each $m\in{\mathbb Z}$, ${\mathcal F}ib(m)$ completely reduces into a direct sum of one trivial module on level $0$, one non-standard module on levels $|m|\leq 2$, and standard highest- and lowest-weight modules on all levels.

This conjecture proposes a uniform decomposition of all level modules in the decomposition of the rank 3 Kac–Moody algebra with respect to ${\mathcal F}ib$. The surrounding discussion establishes complete reducibility in several previously understood cases, but the assertion for all levels is presented as a conjecture.