Progress in Mathematics

2 primary works • 3 total works

Generalized Vertex Algebras and Relative Vertex Operators

by Chongying Dong and James Lepowsky

Published July 1993

The rapidly-evolving theory of vertex operator algebras provides deep insight into many important algebraic structures. Vertex operator algebras can be viewed as "complex analogues" of both Lie algebras and associative algebras. They are mathematically precise counterparts of what are known in physics as chiral algebras, and in particular, they are intimately related to string theory and conformal field theory. Dong and Lepowsky present a generalization of the theory of vertex operator algebras in a systematic way, in three successive stages, all of which involve one-dimensional braid group representations intrinsically in the algebraic structure: First, their notion of "generalized vertex operator algebra" incorporates such structures as Z-algebras, parafermion algebras, and vertex operator superalgebras. Important examples are based on a general construction that they cau "relative vertex operators." Next, what they term "generalized vertex algebras' encompass in addition the algebras of vertex operators associated with rational lattices.
Finally, the most general of the three notions, that of 'abelian intertwining algebra," also iuun-iinates the theory of intertwining operators for certain classes of vertex operator algebras. The monograph is written in an accessible and self-contained way, with detailed proofs and with many examples interwoven through the axiomatic treatment as motivation and applications. It win be useful for research mathematicians and theoretical physicists working in such fields as representation theory and algebraic structures and will provide the basis for a number of graduate courses and seminars on these and related topics.

Introduction to Vertex Operator Algebras and Their Representations

by James Lepowsky and Haisheng Li

Published 19 December 2003

* Introduces the fundamental theory of vertex operator algebras and its basic techniques and examples.

* Begins with a detailed presentation of the theoretical foundations and proceeds to a range of applications.

* Includes a number of new, original results and brings fresh perspective to important works of many other researchers in algebra, lie theory, representation theory, string theory, quantum field theory, and other areas of math and physics.

Functional Analysis on the Eve of the 21st Century

by Simon Gindikin, James Lepowsky, and Robert Wilson

Published 1 December 1995

A four-day conference, "Functional Analysis on the Eve of the Twenty First Century," was held at Rutgers University, New Brunswick, New Jersey, from October 24 to 27, 1993, in honor of the eightieth birthday of Professor Israel Moiseyevich Gelfand. He was born in Krasnye Okna, near Odessa, on September 2, 1913. Israel Gelfand has played a crucial role in the development of functional analysis during the last half-century. His work and his philosophy have in fact helped to shape our understanding of the term "functional analysis" itself, as has the celebrated journal Functional Analysis and Its Applications, which he edited for many years. Functional analysis appeared at the beginning of the century in the classic papers of Hilbert on integral operators. Its crucial aspect was the geometric interpretation of families of functions as infinite-dimensional spaces, and of op erators (particularly differential and integral operators) as infinite-dimensional analogues of matrices, directly leading to the geometrization of spectral theory. This view of functional analysis as infinite-dimensional geometry organically included many facets of nineteenth-century classical analysis, such as power series, Fourier series and integrals, and other integral transforms.