Memoirs of the American Mathematical Society 2002; 119 pp; softcover Volume: 157 ISBN10: 0821828886 ISBN13: 9780821828885 List Price: US$59 Individual Members: US$35.40 Institutional Members: US$47.20 Order Code: MEMO/157/746
 Starting from Borcherds' fake monster Lie algebra we construct a sequence of six generalized KacMoody algebras whose denominator formulas, root systems and all root multiplicities can be described explicitly. The root systems decompose space into convex holes, of finite and affine type, similar to the situation in the case of the Leech lattice. As a corollary, we obtain strong upper bounds for the root multiplicities of a number of hyperbolic Lie algebras, including \(AE_3\). Readership Graduate students and research mathematicians interested in nonassociative rings and algebras. Table of Contents  Introduction
 Generalized KacMoody algebras
 Modular forms
 Lattices and their Thetafunctions
 The proof of Theorem 1.7
 The real simple roots
 Hyperbolic Lie algebras
 Appendix A
 Appendix B
 Bibliography
 Notation
