Frenkel’s bound for the family A_n^{++}

Establish Frenkel’s root-multiplicity bound for the hyperbolic Kac–Moody algebras A_n^{++} beyond the rank-three case A_1^{++}, namely prove that every root α satisfies mult(α) ≤ p_{n-1}(1 − (α,α)/2).

Background

Frenkel’s conjecture proposes that for a symmetric hyperbolic Kac–Moody algebra of rank r, every root α should satisfy mult(α) ≤ p_{r-2}(1 − (α,α)/2). The paper explains that the conjecture is false in full generality because Kac, Moody, and Wakimoto found a counterexample for E_10 = E_8^{++}.

The manuscript proves the conjectured inequality for the rank-three algebra A_1^{++}, but explicitly records that the corresponding problem for the broader family A_n^{++} remained unresolved. Thus, determining whether and in what form Frenkel’s bound holds for the other members of this family is an open problem beyond the case treated in the paper.