Preface

The quantum groups investigated in this book are quantum enveloping al-

gebras defined by their Drinfeld-Jimbo presentation once a symmetrizable

(generalized) Cartan matrix is specified. This presentation is essentially a

(^-deformation or "quantization" of the familiar presentation (by Chevalley

generators and Serre relations) of the universal enveloping algebra of a Kac-

Moody Lie algebra associated with a symmetrizable Cartan matrix. Thus,

one approach to quantum enveloping algebras closely follows the study of

universal enveloping algebras of Lie algebras, the results often amounting to

quantizations of their classical counterparts.

There is a well-known procedure for obtaining symmetrizable Cartan

matrices from finite (possibly valued) graphs. About two decades before the

birth of quantum groups, representations of quivers (i.e., directed graphs)

were introduced and developed as part of both a new approach to the rep-

resentation theory of finite dimensional algebras and a method to deal with

problems in linear algebra. P. Gabriel [118] showed, for example, that if the

underlying graph of a quiver is a (simply laced) Dynkin graph, then the inde-

composable representations correspond naturally to the positive roots of the

finite dimensional complex semisimple Lie algebra associated with the same

Dynkin graph. Over a decade later, V. Kac [170] generalized Gabriel's result

to an arbitrary quiver, obtaining a one-to-one correspondence between the

positive real roots of the associated Lie algebra and certain indecomposable

quiver representations, as well as a one-to-many correspondence from the

positive imaginary roots to the remaining indecomposable representations.

Thus, an essential feature of the structure of a symmetrizable Kac-Moody

Lie algebra — namely, its root space decomposition — has an interpretation

in terms of representations of finite dimensional algebras.

xm