Automorphic correction of the hyperbolic Kac-Moody algebra $E_{10}$

Abstract

In this paper we study automorphic correction of the hyperbolic Kac-Moody algebra $E_{10}$, using the Borcherds product for O(10,2) attached to a weakly holomorphic modular form of weight -4 for $SL_2(\mathbb Z)$. We also clarify some aspects of automorphic correction for Lorentzian Kac-Moody algebras and give heuristic reasons for the expectation that every Lorentzian Kac-Moody algebra has an automorphic correction.