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On the Connection of Certain Lie Algebras with Vertex Algebras

Published online by Cambridge University Press:  06 July 2010

Haisheng Li
Affiliation:
Department of Mathematical Sciences, Rutgers University, Camden, NJ 08102 E-mail address: hli@camden.rutgers.edu
James Lepowsky
Affiliation:
Rutgers University, New Jersey
John McKay
Affiliation:
Concordia University, Montréal
Michael P. Tuite
Affiliation:
National University of Ireland, Galway
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Summary

Abstract

In this paper we survey the connection of certain infinite-dimensional Lie algebras, including twisted and untwisted affine Lie algebras, toroidal Lie algebras and quantum torus Lie algebras, with vertex algebras.

Introduction

Vertex (operator) algebras are a new class of algebraic structures and they have deep connections with numerous fields. In mathematics, vertex algebras have been a vibrant research area. On the other hand, as the algebraic counterpart of chiral algebras, vertex operator algebras together with their representations provide a solid foundation for the study of conformal field theory in physics.

Though vertex algebras are highly non-classical, they have connections with classical algebras such as Lie algebras, associative algebras and groups. In particular, vertex algebras are often constructed and studied by using classical (infinite-dimensional) Lie algebras. For example, those vertex operator algebras associated to (untwisted) affine Kac-Moody Lie algebras (including infinite-dimension Heisenberg Lie algebras) and the Virasoro Lie algebra (cf. [FZ], [DL], [Li1], [LL]) are among the important examples. These two families of vertex operator algebras underline the algebraic study of the physical Wess-Zumino-Novikov-Witten model and the minimal models in conformal field theory, respectively. On the other hand, twisted affine Lie algebras (see [K1]) can be also associated with vertex operator algebras in terms of twisted modules (see [FLM], [Li2]). In the theory of Lie algebras, by generalizing the loop-realization of untwisted affine Lie algebras, one has toroidal Lie algebras, which are perfect central extensions of multi-loop Lie algebras.

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Moonshine - The First Quarter Century and Beyond
Proceedings of a Workshop on the Moonshine Conjectures and Vertex Algebras
, pp. 236 - 254
Publisher: Cambridge University Press
Print publication year: 2010

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