Introduction to Quantum Groups

1. Front Matter

   Pages i-xiv

   PDF

2. The Drinfeld-Jimbo Algebra U

   1. Front Matter

      Pages 1-1

      PDF

3. THE DRINFELD JIMBO ALGERBRA U

   1. The Algebra f

      • George Lusztig

      Pages 2-13

   2. Weyl Group, Root Datum

      • George Lusztig

      Pages 14-18

   3. The Algebra U

      • George Lusztig

      Pages 19-33

   4. The Quasi-\(\mathcal{R}\)-Matrix

      • George Lusztig

      Pages 34-39

   5. The Symmetries \(T^{\prime}_{\rm {i, e}}, \ T^{\prime \prime}_{\rm {i, e}}\) of an Integrable U-Module

      • George Lusztig

      Pages 40-47

   6. Complete Reducibility Theorems

      • George Lusztig

      Pages 48-54

   7. Higher Order Quantum Serre Relations

      • George Lusztig

      Pages 55-60

4. Geometric Realization of F

   1. Front Matter

      Pages 61-62

      PDF

5. GEOMETRIC REALIZATION OF F

   1. Review of the Theory of Perverse Sheaves

      • George Lusztig

      Pages 63-67

   2. Quivers and Perverse Sheaves

      • George Lusztig

      Pages 68-80

   3. Fourier-Deligne Transform

      • George Lusztig

      Pages 81-88

   4. Periodic Functors

      • George Lusztig

      Pages 89-91

   5. Quivers with Automorphisms

      • George Lusztig

      Pages 92-105

   6. The Algebras \(\mathcal{O}^{\prime} {\rm k}\) and k

      • George Lusztig

      Pages 106-112

   7. The Signed Basis of f

      • George Lusztig

      Pages 113-128

6. KASHIWARA’S Operators and Applications

   1. Front Matter

      Pages 129-129

      PDF

7. KASHIWARAS OPERATIONS AND APPLICATIONS

   1. The Algebra \(\mathfrak{U}\)

      • George Lusztig

      Pages 130-131

   2. Kashiwara’s Operators in Rank 1

      • George Lusztig

      Pages 132-141

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.

• Fourier-Deligne transform
• Kac-Moody Lie algebras
• Kashiwara's operators
• Permutation
• Representation theory
• Weyl group
• algebra
• binomial
• braid group relations
• complete reducibility theorems
• homomorphism
• integrable U-module
• perverse sheaves

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)

• MIT, Cambridge, USA

  George Lusztig

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