Introduction to quantum groups

書誌事項

Introduction to quantum groups

George Lusztig

(Progress in mathematics, v. 110)

Birkhäuser, c1993

  • : us
  • : sz

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注記

Includes bibliographical references and indexes

内容説明・目次

巻冊次

: us ISBN 9780817637125

内容説明

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.

目次

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.
巻冊次

: sz ISBN 9783764337124

内容説明

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.

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