Chapter 0

Getting started

This chapter collects together several preliminary topics that will play im-

portant roles throughout the book. In §0.1, we discuss (generalized) Cartan

matrices which arise, for example, in Part 1 in the representation theory of

quivers and in Part 2 in the construction of quantum enveloping algebras.

Two realizations of Cartan matrices — the graph realization and the root

datum realization — are introduced. Since many of the objects encountered

later will be defined by means of a presentation by generators and relations,

§0.2 lays out the general theory, with some important examples presented

in §0.3. Next, §0.4 introduces Gaussian polynomials by showing how they

arise naturally in several counting problems involving finite fields, and §0.5

discusses the theory of "canonical bases" from a very elementary and con-

crete matrix-theoretic point of view. These results will be applied in several

places later in the text, for example, in the discussion of Kazhdan-Lusztig

bases in Chapter 7. Finally, §0.6 provides a brief overview of the theory

of finite dimensional semisimple complex Lie algebras, focusing on the root

systems, with some results extending to Kac-Moody Lie algebras. Strictly

speaking, most of the theory presented there is unnecessary for this book.

However, it not only reveals the origin of the theory of Cartan matrices and

root systems, but also serves as the classical model of the theory of quantum

enveloping algebras, one of the main topics in this book.

0.1. Cartan matrices and their two realizations»

For at least 100 years, mathematicians have known that the classification

of finite dimensional semisimple complex Lie algebras is encoded in certain

integer matrices, the so-called Cartan matrices. By the late 1960s, a much

larger class of Lie algebras — the Kac-Moody Lie algebras — started to play

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http://dx.doi.org/10.1090/surv/150/01