Introduction to Quantum Groups

The quantum groups discussed in this book are the quantized enveloping algebras introduced by Drinfeld and Jimbo in 1985, or variations thereof. It is shown that these algebras have natural integral forms which can be specialised at roots of 1 and yield new objects which include quantum versions of the semisimple groups over fields of positive characteristic. The theory of quantum group has led to a new, extremely rigid structure, in which the objects of the theory are provided with canonical bases with rather remarkable properties. The book contains an extensive treatment of the theory of canonical bases, in the framework of perverse sheaves. The theory developed in the book includes the case of quantum affine enveloping algebras and, more generally, the quantum analogs of the Kac-Moody Lie algebras. This book may be of interest to mathematicians working in representation theory of Lie groups and Lie algebras, knot theorists, and to. theoretical physicists. Since large parts of the book are independent of the theory of perverse sheaves, the book could also be used by graduate students, or as a textbook.