The affine Kac-Moody algebra $A_1 DEGREES{(1)}$ has served as a source of ideas in the representation theory of infinite-dimensional affine Lie algebras. This book develops the calculus of vertex operators to solve the problem of constructing all the standard $A_1 DEGREES{(1)}$-modules in the homogeneou
As the Proceedings of the 1984 Canadian Mathematical Society's Summer Seminar, this book focuses on some advances in the theory of semisimple Lie algebras and some direct outgrowths of that theory. The following papers are of particular interest: an important survey article by R. Block and R. Wilson on restricted simple Lie algebras, a survey of universal enveloping algebras of semisimple Lie algebras by W. Borho, a course on Kac-Moody Lie algebras by I. G. Macdonald with an extensive bibliography of this field by Georgia Benkart, and a course on formal groups by M. Hazewinkel. Because of the expository surveys and courses, the book will be especially useful to graduate students in Lie theory, as well as to researchers in the field.
Discusses the problem of determining the finite-dimensional simple Lie algebras over an algebraically closed field of characteristic $p>7$. This book includes topics such as Lie algebras of prime characteristic, algebraic groups, combinatorics and representation theory, and Kac-Moody and Virasoro algebras.
Using the theory of vertex operator algebras and intertwining operators, we obtain presentations for the principal subspaces of all the standard $widehat{goth{sl}(3)}$-modules. Certain of these presentations had been conjectured and used in work of Calinescu to construct exact sequences leading to the graded dimensions of certain principal subspaces. We prove the conjecture in its full generality for all standard $widehat{goth{sl}(3)}$-modules. We then provide a conjecture for the case of $widehat{goth{sl}(n)}$, $n ge 4$. In addition, we construct completions of certain universal enveloping algebras and provide a natural setting for families of defining relations for the principal subspaces of standard modules for untwisted affine Lie algebras. We also use the theory of vertex operator algebras and intertwining operators, along with conjecturally assumed presentations for certain principal subspaces, to construct exact sequences among principal subspaces of certain standard $widehat{mathfrak{sl}(n)}$-modules, $n ge 3$. As a consequence, we obtain the multigraded dimensions of the principal subspaces $W(k_1Lambda_1 + k_2 Lambda_2)$ and $W(k_{n-2}Lambda_{n-2} + k_{n-1} Lambda_{n-1})$. This generalizes earlier work by Calinescu on principal subspaces of standard $widehat{mathfrak{sl}(3)}$-modules, where similar assumptions were made.
The articles in this book are based on talks given at the international conference 'Lie algebras, vertex operator algebras and their applications'. The focus of the papers is mainly on Lie algebras, quantum groups, vertex operator algebras and their applications to number theory, combinatorics and conformal field theory.
James Lepowsky t The search for symmetry in nature has for a long time provided representation theory with perhaps its chief motivation. According to the standard approach of Lie theory, one looks for infinitesimal symmetry -- Lie algebras of operators or concrete realizations of abstract Lie algebras. A central theme in this volume is the construction of affine Lie algebras using formal differential operators called vertex operators, which originally appeared in the dual-string theory. Since the precise description of vertex operators, in both mathematical and physical settings, requires a fair amount of notation, we do not attempt it in this introduction. Instead we refer the reader to the papers of Mandelstam, Goddard-Olive, Lepowsky-Wilson and Frenkel-Lepowsky-Meurman. We have tried to maintain consistency of terminology and to some extent notation in the articles herein. To help the reader we shall review some of the terminology. We also thought it might be useful to supplement an earlier fairly detailed exposition of ours [37] with a brief historical account of vertex operators in mathematics and their connection with affine algebras. Since we were involved in the development of the subject, the reader should be advised that what follows reflects our own understanding. For another view, see [29].1 t Partially supported by the National Science Foundation through the Mathematical Sciences Research Institute and NSF Grant MCS 83-01664. 1 We would like to thank Igor Frenkel for his valuable comments on the first draft of this introduction.
