Introduction to Quantum Groups

According to Drinfeld, a quantum group is the same as a Hopf algebra. This includes as special cases, the algebra of regular functions on an algebraic group and the enveloping algebra of a semisimple Lie algebra. The qu- tum groups discussed in this book are the quantized enveloping algebras introduced by Drinfeld and Jimbo in 1985, or variations thereof. Although such quantum groups appeared in connection with problems in statistical mechanics and are closely related to conformal field theory and knot theory, we will regard them purely as a new development in Lie theory. Their place in Lie theory is as follows. Among Lie groups and Lie algebras (whose theory was initiated by Lie more than a hundred years ago) the most important and interesting ones are the semisimple ones. They were classified by E. Cartan and Killing around 1890 and are quite central in today's mathematics. The work of Chevalley in the 1950s showed that semisimple groups can be defined over arbitrary fields (including finite ones) and even over integers. Although semisimple Lie algebras cannot be deformed in a non-trivial way, the work of Drinfeld and Jimbo showed that their enveloping (Hopf) algebras admit a rather interesting deformation depending on a parameter v. These are the quantized enveloping algebras of Drinfeld and Jimbo. The classical enveloping algebras could be obtained from them for v -» 1.

List of contents

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.

Summary

According to Drinfeld, a quantum group is the same as a Hopf algebra. This includes as special cases, the algebra of regular functions on an algebraic group and the enveloping algebra of a semisimple Lie algebra. The qu- tum groups discussed in this book are the quantized enveloping algebras introduced by Drinfeld and Jimbo in 1985, or variations thereof. Although such quantum groups appeared in connection with problems in statistical mechanics and are closely related to conformal field theory and knot theory, we will regard them purely as a new development in Lie theory. Their place in Lie theory is as follows. Among Lie groups and Lie algebras (whose theory was initiated by Lie more than a hundred years ago) the most important and interesting ones are the semisimple ones. They were classified by E. Cartan and Killing around 1890 and are quite central in today's mathematics. The work of Chevalley in the 1950s showed that semisimple groups can be defined over arbitrary fields (including finite ones) and even over integers. Although semisimple Lie algebras cannot be deformed in a non-trivial way, the work of Drinfeld and Jimbo showed that their enveloping (Hopf) algebras admit a rather interesting deformation depending on a parameter v. These are the quantized enveloping algebras of Drinfeld and Jimbo. The classical enveloping algebras could be obtained from them for v —» 1.

Additional text

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)

Report

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)

Product details

• Authors: George Lusztig
• Publisher: Springer, Basel
• Languages: English
• Product format: Paperback / Softback
• Released: 01.01.2011
• EAN: 9780817647162
• ISBN: 978-0-8176-4716-2
• No. of pages: 352
• Dimensions: 156 mm x 22 mm x 234 mm
• Weight: 544 g
• Illustrations: XIV, 352 p.
• Series: Progress in Mathematics; Modern Birkhäuser Classics; Progress in Mathematics; Modern Birkhauser Classics; Modern Birkhäuser Classics
• Subject: Natural sciences, medicine, IT, technology > Mathematics > Arithmetic, algebra