This work is motivated by and develops connections between several branches of mathematics and physics--the theories of Lie algebras, finite groups and modular functions in mathematics, and string theory in physics. The first part of the book presents a new mathematical theory of vertex operator algebras, the algebraic counterpart of two-dimensional holomorphic conformal quantum field theory. The remaining part constructs the Monster finite simple group as the automorphism group of a very special vertex operator algebra, called the "moonshine module" because of its relevance to "monstrous moonshine."

lgrsnf/Frenkel I., Lepowsky J., Meurman A. Vertex operator algebras and the monster (AP, 1988)(ISBN 0122670655)(T)(O)(563s)_MAa_.djvu

Frenkel, Igor, Lepowsky, James, Meurman, Arne

Morgan Kaufmann Publishers

Brooks/Cole

Pure and applied mathematics ;, v. 134, Pure and applied mathematics (Academic Press) ;, 134., Boston, Massachusetts, 1988

Pure and applied mathematics (Academic Press), Place of publication not identified, 1989

Pure and applied mathematics (Academic Press), v. 134, Boston, 1988

Pure a. appl. mathematics -- Vol. 003134, Boston etc, Unknown, 1988

United States, United States of America

Elsevier Ltd., Boston, 1988

March 28, 1989

Bibliography: p. 483-492.
Includes index.

Библиогр.: с. 483-492
Указ.

РГБ

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Front Cover
Vertex Operator Algebras and the Monster
Copyright Page
Contents
Preface
Introduction
Notational Conventions
Chapter 1. Lie Algebras
1.1. Algebras
1.2. Modules
1.3. Algebra Constructions
1.4. Module Constructions
1.5. Induced Modules
1.6. Affine Lie Algebras
1.7. Heisenberg Algebras
1.8. Contravariant Forms
1.9. The Virasoro Algebra
1.10. Graded Dimension
Chapter 2. Formal Calculus
2.1. Formal Series
2.2. Derivations
2.3. Affine Lie Algebras via Formal Variables
Chapter 3. Realizations of l(2) by Twisted Vertex Operators
3.1. The Affine Lie Algebra SI(2)
3.3. Normal Ordering
3.4. Some Commutators
3.5. Irreducible Representations of SI(2)[O2]
Chapter 4. Realizations of SI(2) by Untwisted Vertex Operators
4.2. Normal Ordering
4.3. Some Commutators
4.4. Irreducible Representations of SI(2) and SI(2)[01]
4.5. Isomorphism of Two Constructions
Chapter 5. Central Extensions
5.1. 2-Cocycles
5.2. Commutator Maps
5.3. Extraspecial 2-Groups
5.4. Automorphisms of Central Extensions
5.5 . Representations of Central Extensions
Chapter 6. The Simple Lie Algebras An, Dn, En
6.1. Lattices
6.2. A Class of Lie Algebras
6.3. The Cases An, Dn, En
6.4. A Group of Automorphisms of g
Chapter 7. Vertex Operator Realizations of Ân, Dn, Ên
7.2. Construction of Ân, Dn, Ên
7.4. Construction of Ân[0], Dn[0], Ê[0]
Chapter 8. General Theory of Untwisted Vertex Operators
8.1. Expansions of Zero
8.2. Exponentials of Derivations
8.3. Projective Changes of Variable and Higher Derivatives of Composite Functions
8.4. Commutators of Untwisted Vertex Operators
8.5. General Vertex Operators
8.6. Commutators of General Vertex Operators
8.7. The Virasoro Algebra Revisited
8.8. The Jacobi Identity
8.9. Cross-brackets and Commutative Affinization
8.10. Vertex Operator Algebras and the Rationality, Commutativity and Associativity Properties
Chapter 9. General Theory of Twisted Vertex Operators
9.1. Commutators of Twisted Vertex Operators
9.2. General Twisted Vertex Operators
9.3. Commutators of General Twisted Vertex Operators
9.4. The Virasoro Algebra: Twisted Construction Revisited
9.5. The Jacobi Identity and Cross-Brackets: Twisted Case
Chapter 10. The Moonshine Module
10.1. The Golay Code
10.2. The Leech Lattice
10.3. The Moonshine Module Vh and the Griess Algebra B
10.4. The Group C and Its Actions on Vh and on B
10.5. The Graded Character of the C-Module Vh
Chapter 11. Triality
11.1. The Setting
11.5. Summary
Chapter 12. The Main Theorem
12.1 The Main Setting
12.3. The Monster M and the Statement of the Main Theorem
12.4. The M-Invariant Q-Form VkQ
12.5. The M-Invariant Positive Definite Hermitian Form
Chapter 13. Completion of the Proof
13.1. Reduction to Two Lemmas
13.2. Groups Acting on WL : Proof of Lemma 13.1.2
13.3. Action of D6/4 on W£
13.4. Groups Acting on W£
13.5. Some Group Cohomology
13.6. A Splitting of a Sequence: Proof of Lemma 13.4.7
Appendix: Complex Realization of Vertex Operator Algebras
A.1. Linear Algebra of Infinite Direct Sums
A.2. The Vertex Operator Algebra VL
A.3. Relation to the Formal Variable Approach
Bibliography
List of Frequently Used Symbols
Index
Pure and Applied Mathematics

2024-07-28