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. The theory of quantum groups has led to a new, extremely rigid structure, in which the objects of the theory are provided with canonical basis with rather remarkable properties. This book will be of interest to mathematicians working in the representation theory of Lie groups and Lie algebras, knot theorists and to theoretical physicists and graduate students. Since large parts of the book are independent of the theory of perverse sheaves, the book could also be used as a text book.

From the reviews:"There is no doubt that this volume is a very remarkable piece of work...Its appearance represents a landmark in the mathematical literature."—Bulletin of the London Mathematical Society"This book is an important contribution to the field and can be recommended especially to mathematicians working in the field."—EMS Newsletter"The present book gives a very efficient presentation of an important part of quantum group theory. It is a valuable contribution to the literature."—Mededelingen van het Wiskundig"Lusztig's book is very well written and seems to be flawless...Obviously, this will be the standard reference book for the material presented and anyone interested in the Drinfeld–Jimbo algebras will have to study it very carefully."—ZAA"[T]his book is much more than an 'introduction to quantum groups.' It contains a wealth of material. In addition to the many important results (of which several are new–at least in the generality presented here), there are plenty of useful calculations (commutator formulas, generalized quantum Serre relations, etc.)."—Zentralblatt MATH“George Lusztig lays out the large scale structure of the discussion that follows in the 348 pages of his Introduction to Quantum Groups. … A significant and important work. … it’s terrific stuff, elegant and deep, and Lusztig presents it very well indeed, of course.” (Michael Berg, The Mathematical Association of America, January, 2011)