Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
6414608 | Journal of Algebra | 2014 | 24 Pages |
In this paper we study rank 2 symmetric hyperbolic Kac-Moody algebras H(a) with the Cartan matrices (2âaâa2), 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(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 H(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.