P r e f a c e
These volumes contain papers based on lectures presented at the confer-
ence on "Algebraic Groups and Their Generalizations" held at Pennsylvania
State University from July 6-26, 1991 and organized by Igor Frenkel, William
Haboush (Chair), Jens Jantzen, and Brian Parshall. A response to the remark-
able proliferation of Lie theory in the last decade and a half, this conference
reflected both the diversification of technique in the classical theory and the
beginnings of the study of new objects such as quantum groups, vertex oper-
ator algebras, and various kinds of infinite-dimensional groups and algebras,
inspired by new work in mathematical physics and quantum field theory. It
is hoped that the papers in these volumes will provide the interested reader
with an introduction to these astonishing new mathematical worlds.
The first volume focuses on classical methods, while the second centers on
quantum and infinite-dimensional methods. Each section begins with various
expositions and then turns to papers on previously unknown results.
As construed by the editors, classical methods include the geometry of
Schubert varieties and homogeneous spaces, Frobenius splitting techniques,
the apparatus of Kazhdan-Lusztig theory including uses of intersection ho-
mology, the study of certain categories of representations based on highest
weight modules, and invariant theory.
Likewise, the notions of quantum and infinite-dimensional theory have
been broadly defined to include various notions of quantum groups and
quantized enveloping algebras, crystal bases, vertex operator algebras, and
infinite-dimensional objects and topics such as braidings and the equations
of Yang-Baxter type and their generalizations. We hope that these volumes
represent a useful addition to the basic literature in these areas.
The editors particularly wish to acknowledge the generous assistance of
Catherine Chevalley who made possible the publication of the article by
Claude Chevalley in this volume. The belated appearance of this manuscript,
which has influenced so many mathematicians, makes its special insights,
both historical and mathematical, available to the mathematics community.
For this we thank Ms. Chevalley.
William J. Haboush
Brian J. Parshall