**Memoirs of the American Mathematical Society**

2002;
119 pp;
Softcover

MSC: Primary 17;

Print ISBN: 978-0-8218-2888-5

Product Code: MEMO/157/746

List Price: $62.00

AMS Member Price: $37.20

MAA Member Price: $55.80

**Electronic ISBN: 978-1-4704-0339-3
Product Code: MEMO/157/746.E**

List Price: $62.00

AMS Member Price: $37.20

MAA Member Price: $55.80

# Some Generalized Kac-Moody Algebras with Known Root Multiplicities

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*Peter Niemann*

Starting from Borcherds' fake monster Lie algebra we construct a sequence of six generalized Kac-Moody 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

# Table of Contents

## Some Generalized Kac-Moody Algebras with Known Root Multiplicities

- Contents vii8 free
- Introduction 112 free
- Chapter 1. Generalized Kac-Moody Algebras 516 free
- Chapter 2. Modular Forms 2940
- Chapter 3. Lattices and their Theta-Functions 3445
- Chapter 4. The Proof of Theorem 1.7 3849
- Chapter 5. The Real Simple Roots 5465
- Chapter 6. Hyperbolic Lie Algebras 7283
- Appendix A 93104
- Appendix B 110121
- Bibliography 116127
- Notation 118129