Rank 2 symmetric hyperbolic Kac-Moody algebras and Hilbert modular forms

In this paper we study rank 2 symmetric hyperbolic Kac Moody algebras H/(a) with the Cartan matrices ((2)(-2) (-a)(2)), a >= 3, and their automorphic correction in terms of Hilbert modular forms. We associate a family of H(a)'s to the quadratic field Q(root p) for each odd prime p and show that there exists a chain of embeddings in each family. When p = 5, 13, 17, we show that the first 71(a) in each family, i.e. H(3), H(11), H(66), is contained in a generalized Kac Moody superalgebra whose denominator function is a Hilbert modular form given by a Borcherds product. Hence, our results provide automorphic correction for those H(a)'s. We also compute asymptotic formulas for the root multiplicities of the generalized Kac-Moody superalgebras using the fact that the exponents in the Borcherds products are Fourier coefficients of weakly holomorphic modular forms of weight 0. (c) 2014 Elsevier Inc. All rights reserved.