A proof of Frenkel's bound for the hyperbolic Kac-Moody Lie Algebra A_1^++

Abstract

We give a computer-assisted proof of Frenkel's root-multiplicity bound for the rank-three hyperbolic Kac--Moody Lie algebra $A_1^{++}$. For every root $α$, we prove $\dim g_α\le p(1-(α,α)/2)$, where $p$ is the ordinary partition function. The proof combines exact affine characters with a coefficientwise logarithmic majorant derived from parabolic homology. After Weyl reduction, analytic estimates establish the inequality on infinite regions of large depth or large level. The remaining finite region is verified by computer. Together with the known low-level formulas, the assembly of these estimates establish the conjecture for every root of $A_1^{++}$.