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.
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