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Book DetailsContemporary MathematicsVolume: 121; 1991; 146 ppMSC: Primary 17; 81;
The theory of vertex operator algebras is a remarkably rich new mathematical field which captures the algebraic content of conformal field theory in physics. Ideas leading up to this theory appeared in physics as part of statistical mechanics and string theory. In mathematics, the axiomatic definitions crystallized in the work of Borcherds and in Vertex Operator Algebras and the Monster, by Frenkel, Lepowsky, and Meurman. The structure of monodromies of intertwining operators for modules of vertex operator algebras yields braid group representations and leads to natural generalizations of vertex operator algebras, such as superalgebras and paraalgebras. Many examples of vertex operator algebras and their generalizations are related to constructions in classical representation theory and shed new light on the classical theory.
This book accomplishes several goals. The authors provide an explicit spinor construction, using only Clifford algebras, of a vertex operator superalgebra structure on the direct sum of the basic and vector modules for the affine KacMoody algebra \(D^{(1)}_n\). They also review and extend Chevalley's spinor construction of the 24dimensional commutative nonassociative algebraic structure and triality on the direct sum of the three 8dimensional \(D_4\)modules. Vertex operator paraalgebras, introduced and developed independently in this book and by Dong and Lepowsky, are related to onedimensional representations of the braid group. The authors also provide a unified approach to the Chevalley, Griess, and \(E_8\) algebras and explain some of their similarities. A third goal is to provide a purely spinor construction of the exceptional affine Lie algebra \(E^{(1)}_8\), a natural continuation of previous work on spinor and oscillator constructions of the classical affine Lie algebras. These constructions should easily extend to include the rest of the exceptional affine Lie algebras. The final objective is to develop an inductive technique of construction which could be applied to the Monster vertex operator algebra.
Directed at mathematicians and physicists, this book should be accessible to graduate students with some background in finitedimensional Lie algebras and their representations. Although some experience with affine KacMoody algebras would be useful, a summary of the relevant parts of that theory is included. This book shows how the concepts and techniques of Lie theory can be generalized to yield the algebraic structures associated with conformal field theory. The careful reader will also gain a detailed knowledge of how the spinor construction of classical triality lifts to the affine algebras and plays an important role in a spinor construction of vertex operator algebras, modules, and intertwining operators with nontrivial monodromies. 
Table of Contents

Chapters

Chapter 0: Introduction

Chapter 1: Summary

Chapter 2: Affine Algebras and Representations

Chapter 3: Spinor Construction of Vertex Operator Superalgebras

Chapter 4: Spinor Construction of the Chevalley Algebra and Triality for $D_4$

Chapter 5: Spinor Construction of Triality for $D^{(1)}_4$

Chapter 6: Spinor Construction of a Vertex Operator Paraalgebra for $D^{(1)}_4$

Chapter 7: Spinor Construction of $E_8$

Chapter 8: Spinor Construction of Vertex Operator Algebras for $E^{(1)}_8$

References


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A successful attempt to describe a common background of recent investigations.
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The theory of vertex operator algebras is a remarkably rich new mathematical field which captures the algebraic content of conformal field theory in physics. Ideas leading up to this theory appeared in physics as part of statistical mechanics and string theory. In mathematics, the axiomatic definitions crystallized in the work of Borcherds and in Vertex Operator Algebras and the Monster, by Frenkel, Lepowsky, and Meurman. The structure of monodromies of intertwining operators for modules of vertex operator algebras yields braid group representations and leads to natural generalizations of vertex operator algebras, such as superalgebras and paraalgebras. Many examples of vertex operator algebras and their generalizations are related to constructions in classical representation theory and shed new light on the classical theory.
This book accomplishes several goals. The authors provide an explicit spinor construction, using only Clifford algebras, of a vertex operator superalgebra structure on the direct sum of the basic and vector modules for the affine KacMoody algebra \(D^{(1)}_n\). They also review and extend Chevalley's spinor construction of the 24dimensional commutative nonassociative algebraic structure and triality on the direct sum of the three 8dimensional \(D_4\)modules. Vertex operator paraalgebras, introduced and developed independently in this book and by Dong and Lepowsky, are related to onedimensional representations of the braid group. The authors also provide a unified approach to the Chevalley, Griess, and \(E_8\) algebras and explain some of their similarities. A third goal is to provide a purely spinor construction of the exceptional affine Lie algebra \(E^{(1)}_8\), a natural continuation of previous work on spinor and oscillator constructions of the classical affine Lie algebras. These constructions should easily extend to include the rest of the exceptional affine Lie algebras. The final objective is to develop an inductive technique of construction which could be applied to the Monster vertex operator algebra.
Directed at mathematicians and physicists, this book should be accessible to graduate students with some background in finitedimensional Lie algebras and their representations. Although some experience with affine KacMoody algebras would be useful, a summary of the relevant parts of that theory is included. This book shows how the concepts and techniques of Lie theory can be generalized to yield the algebraic structures associated with conformal field theory. The careful reader will also gain a detailed knowledge of how the spinor construction of classical triality lifts to the affine algebras and plays an important role in a spinor construction of vertex operator algebras, modules, and intertwining operators with nontrivial monodromies.

Chapters

Chapter 0: Introduction

Chapter 1: Summary

Chapter 2: Affine Algebras and Representations

Chapter 3: Spinor Construction of Vertex Operator Superalgebras

Chapter 4: Spinor Construction of the Chevalley Algebra and Triality for $D_4$

Chapter 5: Spinor Construction of Triality for $D^{(1)}_4$

Chapter 6: Spinor Construction of a Vertex Operator Paraalgebra for $D^{(1)}_4$

Chapter 7: Spinor Construction of $E_8$

Chapter 8: Spinor Construction of Vertex Operator Algebras for $E^{(1)}_8$

References

A successful attempt to describe a common background of recent investigations.
Zentralblatt MATH