Introduction to lie algebras and representation theory

By Vyjayanthi Chari and Andrew Pressley: 651 pp., £22.95 (US$34.95), isbn 0 521 55884 0 (Cambridge University Press, 1994).

A guide to quantum groups

Vertex Algebras and Algebraic Curves

This paper gives an analogue of Ag(V) theory for a vertex operator superalgebra V and an automorphism g of finite order. The relation between the g-twisted V-modules and Ag(V)-modules is established. It is proved that if V is g-rational, then Ag(V) is finite-dimensional semi-simple associative algebra and there are only finitely many irreducible g-twisted V-modules.

Twisted representations of vertex operator algebras

Local systems of vertex operators, vertex superalgebras and modules

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.

Introduction to Vertex Operator Algebras and Their Representations

The goal of the present paper is to provide a mathematically rigorous foundation to certain aspects of the theory of rational orbifold models in conformal field theory, in other words the theory of rational vertex operator algebras and their automorphisms.

Haisheng Li

Papers

Introduction to Vertex Operator Algebras and Their Representations

Twisted representations of vertex operator algebras

Local systems of vertex operators, vertex superalgebras and modules

Modular-Invariance of Trace Functions¶in Orbifold Theory and Generalized Moonshine

Regularity of Rational Vertex Operator Algebras