Springer, 2010. — 352 p. — ISBN 0817647163.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… 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. Table of contents The Algebra f Weyl Group, Root Datum The Algebra U The Quasi- R-Matrix The Symmetries T′i,e, T′′i, E of an Integrable U-Module Complete Reducibility Theorems Higher Order Quantum Serre Relations Review of the Theory of Perverse Sheaves Quivers and Perverse Sheaves Fourier-Deligne Transform Periodic Functors Quivers with Automorphisms The Algebras O′k and k The Signed Basis of f The Algebra U Kashiwara’s Operators in Rank 1 Applications Study of the Operators ~Fi, ~Ei onΛλ Inner Product on Λ Bases at ∞ Cartan Data of Finite Type Positivity of the Action of F i , E i in the Simply-Laced Case The Algebra ˙U Canonical Bases in Certain Tensor Products The Canonical Basis ˙B of ˙U Inner Product on U Based Modules Bases for Coinvariants and Cyclic Permutations A Refinement of the Peter-Weyl Theorem The Canonical Topological Basis of (U − ⨂ U+) The Algebra R˙U Commutativity Isomorphism Relation with Kac-Moody Lie Algebras Gaussian Binomial Coefficients at Roots of 1 The Quantum Frobenius Homomorphism The Algebras Rf, Ru The Symmetries T’ieT′i,e of U Symmetries and Inner Product on f Braid Group Relations Symmetries and U+ Integrality Properties of the Symmetries The ADE Case
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