THE DRINFELD JIMBO ALGERBRA U -- The Algebra f -- Weyl Group, Root Datum -- The Algebra U -- The Quasi--Matrix -- The Symmetries of an Integrable U-Module -- Complete Reducibility Theorems -- Higher Order Quantum Serre Relations -- GEOMETRIC REALIZATION OF F -- Review of the Theory of Perverse Sheaves -- Quivers and Perverse Sheaves -- Fourier-Deligne Transform -- Periodic Functors -- Quivers with Automorphisms -- The Algebras and k -- The Signed Basis of f -- KASHIWARAS OPERATIONS AND APPLICATIONS -- The Algebra -- Kashiwara’s Operators in Rank 1 -- Applications -- Study of the Operators -- Inner Product on -- Bases at ? -- Cartan Data of Finite Type -- Positivity of the Action of Fi, Ei in the Simply-Laced Case -- CANONICAL BASIS OF U -- The Algebra -- Canonical Bases in Certain Tensor Products -- The Canonical Basis -- Inner Product on -- Based Modules -- Bases for Coinvariants and Cyclic Permutations -- A Refinement of the Peter-Weyl Theorem -- The Canonical Topological Basis of -- CHANGE OF RINGS -- The Algebra -- Commutativity Isomorphism -- Relation with Kac-Moody Lie Algebras -- Gaussian Binomial Coefficients at Roots of 1 -- The Quantum Frobenius Homomorphism -- The Algebras -- BRAID GROUP ACTION -- The Symmetries of U -- Symmetries and Inner Product on f -- Braid Group Relations -- Symmetries and U+ -- Integrality Properties of the Symmetries -- The ADE Case.

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 that can be specialized at roots of 1 and yield new objects, which include quantum versions of the semi-simple groups over fields of positive characteristic. The theory of quantum groups has led to a new, extremely rigid structure, in which the objects of the theory are provided with canonical bases having rather remarkable properties. This 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. Introduction to Quantum Groups will be of interest to mathematicians working in the representation theory of Lie groups and Lie algebras, knot theorists, theoretical physicists, and graduate students. Since large parts of the book are independent of the theory of perverse sheaves, the work may also be used as a textbook. **************************************** 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