Introduction to Quantum Groups

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.