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" Introduction to Vertex Operator Algebras and Their Representations "
by James Lepowsky, Haisheng Li.
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BL
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573362
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b402581
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| Main Entry
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Lepowsky, James.
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| Title & Author
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Introduction to Vertex Operator Algebras and Their Representations\ by James Lepowsky, Haisheng Li.
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| Publication Statement
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Boston, MA :: Birkhäuser Boston :: Imprint: Birkhäuser,, 2004.
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| Series Statement
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Progress in Mathematics ;; 227
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| ISBN
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9780817681869
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: 9781461264804
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| Contents
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1 Introduction -- 1.1 Motivation -- 1.2 Example of a vertex operator -- 1.3 The notion of vertex operator algebra -- 1.4 Simplification of the definition -- 1.5 Representations and modules -- 1.6 Construction of families of examples -- 1.7 Some further developments -- 2 Formal Calculus -- 2.1 Formal series and the formal delta function -- 2.2 Derivations and the formal Taylor Theorem -- 2.3 Expansions of zero and applications -- 3 Vertex Operator Algebras: The Axiomatic Basics -- 3.1 Definitions and some fundamental properties -- 3.2 Commutativity properties -- 3.3 Associativity properties -- 3.4 The Jacobi identity from commutativity and associativity -- 3.5 The Jacobi identity from commutativity -- 3.6 The Jacobi identity from skew symmetry and associativity -- 3.7 S3-symmetry of the Jacobi identity -- 3.8 The iterate formula and normal-ordered products -- 3.9 Further elementary notions -- 3.10 Weak nilpotence and nilpotence -- 3.11 Centralizers and the center -- 3.12 Direct product and tensor product vertex algebras -- 4 Modules -- 4.1 Definition and some consequences -- 4.2 Commutativity properties -- 4.3 Associativity properties -- 4.4 The Jacobi identity as a consequence of associativity and commutativity properties -- 4.5 Further elementary notions -- 4.6 Tensor product modules for tensor product vertex algebras -- 4.7 Vacuum-like vectors -- 4.8 Adjoining a module to a vertex algebra -- 5 Representations of Vertex Algebras and the Construction of Vertex Algebras and Modules -- 5.1 Weak vertex operators -- 5.2 The action of weak vertex operators on the space of weak vertex operators -- 5.3 The canonical weak vertex algebra ?(W) and the equivalence between modules and representations -- 5.4 Subalgebras of ?(W) -- 5.5 Local subalgebras and vertex subalgebras of ?(W) -- 5.6 Vertex subalgebras of ?(W) associated with the Virasoro algebra -- 5.7 General construction theorems for vertex algebras and modules -- 6 Construction of Families of Vertex Operator Algebras and Modules -- 6.1 Vertex operator algebras and modules associated to the Virasoro algebra -- 6.2 Vertex operator algebras and modules associated to affine Lie algebras -- 6.3 Vertex operator algebras and modules associated to Heisenberg algebras -- 6.4 Vertex operator algebras and modules associated to even lattices-the setting -- 6.5 Vertex operator algebras and modules associated to even lattices-the main results -- 6.6 Classification of the irreducible L?(?, O)-modules for g finite-dimensional simple and ? a positive integer -- References.
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| Abstract
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Vertexoperatoralgebra theory is a new area of mathematics. It has been an exciting and ever-growing subject from the beginning, starting even before R.Borcherds introduced theprecise mathematical notion of"vertex algebra" inthe 1980s [BI].Having developed in conjunction with string theory in theoretical physics and with the theory of "mon strous moonshine" and infinite-dimensional Lie algebra theory in mathematics, vertex (operator) algebra theory is qualitatively different from traditional algebraic theories, reflecting the "nonclassical" nature of string theory and of monstrous moonshine. The theory has revealed new perspectives that were unavailable without it, and continues to do so. "Monstrous moonshine" began as an astonishing set ofconjectures relating theMon ster finite simple group to the theory of modular functions in number theory.As is now known, vertex operator algebra theory is a foundational pillarof monstrous moonshine. With the theory available, one can formulate and try to solve new problems that have far-reaching implications in a wide range of fields that had not previously been thought of as being related.
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Mathematics.
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Algebra.
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Topological Groups.
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Operator theory.
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Li, Haisheng.
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SpringerLink (Online service)
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