• Mathematics

Quantum Groups


Author: Christian Kassel
Publisher: Springer Science & Business Media
ISBN: 1461207835
Category: Mathematics
Page: 534
View: 5143
Here is an introduction to the theory of quantum groups with emphasis on the spectacular connections with knot theory and Drinfeld's recent fundamental contributions. It presents the quantum groups attached to SL2 as well as the basic concepts of the theory of Hopf algebras. Coverage also focuses on Hopf algebras that produce solutions of the Yang-Baxter equation and provides an account of Drinfeld's elegant treatment of the monodromy of the Knizhnik-Zamolodchikov equations.

    • Mathematics

Lectures on Quantum Groups


Author: Jens Carsten Jantzen
Publisher: American Mathematical Soc.
ISBN: 9780821872345
Category: Mathematics
Page: 266
View: 3388
Starting with the quantum analog of sl2, the author carefully leads the reader through all the details necessary for full understanding of the subject, particularly emphasizing similarities and differences with the classical theory. The final chapters of the book describe the Kashiwara-Lusztig theory of so-called crystal (or canonical) bases in representations of complex semisimple Lie algebra.

    • Science

Quantum Theory for Mathematicians


Author: Brian C. Hall
Publisher: Springer Science & Business Media
ISBN: 1461471168
Category: Science
Page: 554
View: 7609
Although ideas from quantum physics play an important role in many parts of modern mathematics, there are few books about quantum mechanics aimed at mathematicians. This book introduces the main ideas of quantum mechanics in language familiar to mathematicians. Readers with little prior exposure to physics will enjoy the book's conversational tone as they delve into such topics as the Hilbert space approach to quantum theory; the Schrödinger equation in one space dimension; the Spectral Theorem for bounded and unbounded self-adjoint operators; the Stone–von Neumann Theorem; the Wentzel–Kramers–Brillouin approximation; the role of Lie groups and Lie algebras in quantum mechanics; and the path-integral approach to quantum mechanics. The numerous exercises at the end of each chapter make the book suitable for both graduate courses and independent study. Most of the text is accessible to graduate students in mathematics who have had a first course in real analysis, covering the basics of L2 spaces and Hilbert spaces. The final chapters introduce readers who are familiar with the theory of manifolds to more advanced topics, including geometric quantization.

    • Mathematics

A Guide to Quantum Groups


Author: Vyjayanthi Chari,Andrew N. Pressley
Publisher: Cambridge University Press
ISBN: 9780521558846
Category: Mathematics
Page: 651
View: 4291
Since they first arose in the 1970s and early 1980s, quantum groups have proved to be of great interest to mathematicians and theoretical physicists. This book gives a comprehensive view of quantum groups and their applications. The authors build on a self-contained account of the foundations of the subject and go on to treat the more advanced aspects concisely and with detailed references to the literature. Researchers in mathematics and theoretical physics will enjoy this book.

    • Mathematics

Introduction to Quantum Groups and Crystal Bases


Author: Jin Hong,Seok-Jin Kang
Publisher: American Mathematical Soc.
ISBN: 0821828746
Category: Mathematics
Page: 307
View: 7472
The notion of a ``quantum group'' was introduced by V.G. Dinfeld and M. Jimbo, independently, in their study of the quantum Yang-Baxter equation arising from 2-dimensional solvable lattice models. Quantum groups are certain families of Hopf algebras that are deformations of universal enveloping algebras of Kac-Moody algebras. And over the past 20 years, they have turned out to be the fundamental algebraic structure behind many branches of mathematics and mathematical physics, such as solvable lattice models in statistical mechanics, topological invariant theory of links and knots, representation theory of Kac-Moody algebras, representation theory of algebraic structures, topological quantum field theory, geometric representation theory, and $C^*$-algebras. In particular, the theory of ``crystal bases'' or ``canonical bases'' developed independently by M. Kashiwara and G. Lusztig provides a powerful combinatorial and geometric tool to study the representations of quantum groups. The purpose of this book is to provide an elementary introduction to the theory of quantum groups and crystal bases, focusing on the combinatorial aspects of the theory.

    • Mathematics

An Invitation to Quantum Groups and Duality

From Hopf Algebras to Multiplicative Unitaries and Beyond
Author: Thomas Timmermann
Publisher: European Mathematical Society
ISBN: 9783037190432
Category: Mathematics
Page: 407
View: 7759
This book provides an introduction to the theory of quantum groups with emphasis on their duality and on the setting of operator algebras. The book is addressed to graduate students and non-experts from other fields. Only basic knowledge of (multi-)linear algebra is required for the first part, while the second and third part assume some familiarity with Hilbert spaces, CÝsuperscript *¨-algebras, and von Neumann algebras.

    • Science

An Introduction to Quantum Theory


Author: Keith Hannabuss
Publisher: Clarendon Press
ISBN: 9780191588730
Category: Science
Page: 394
View: 7664
This book provides an introduction to quantum theory primarily for students of mathematics. Although the approach is mainly traditional the discussion exploits ideas of linear algebra, and points out some of the mathematical subtleties of the theory. Amongst the less traditional topics are Bell's inequalities, coherent and squeezed states, and introductions to group representation theory. Later chapters discuss relativistic wave equations and elementary particle symmetries from a group theoretical standpoint rather than the customary Lie algebraic approach. This book is intended for the later years of an undergraduate course or for graduates. It assumes a knowledge of basic linear algebra and elementary group theory, though for convenience these are also summarized in an appendix.

    • Mathematics

Foundations of Quantum Group Theory


Author: Shahn Majid
Publisher: Cambridge University Press
ISBN: 9780521648684
Category: Mathematics
Page: 640
View: 2521
Now in paperback, this is a graduate level text for theoretical physicists and mathematicians which systematically lays out the foundations for the subject of Quantum Groups in a clear and accessible way. The topic is developed in a logical manner with quantum groups (Hopf Algebras) treated as mathematical objects in their own right. After formal definitions and basic theory, the book goes on to cover such topics as quantum enveloping algebras, matrix quantum groups, combinatorics, cross products of various kinds, the quantum double, the semiclassical theory of Poisson-Lie groups, the representation theory, braided groups and applications to q-deformed physics. Explicit proofs and many examples will allow the reader quickly to pick up the techniques needed for working in this exciting new field.

    • Mathematics

A Quantum Groups Primer


Author: Shahn Majid
Publisher: Cambridge University Press
ISBN: 9780521010412
Category: Mathematics
Page: 169
View: 1096
Self-contained introduction to quantum groups as algebraic objects, suitable as a textbook for graduate courses.

    • Science

Quantum Groups in Two-Dimensional Physics


Author: Cisar Gómez,Martm Ruiz-Altaba,German Sierra
Publisher: Cambridge University Press
ISBN: 9780521020046
Category: Science
Page: 476
View: 9045
A 1996 introduction to integrability and conformal field theory in two dimensions using quantum groups.

    • Mathematics

Introduction to Quantum Groups


Author: George Lusztig
Publisher: Springer Science & Business Media
ISBN: 9780817647179
Category: Mathematics
Page: 352
View: 528
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.

    • Mathematics

A Course in the Theory of Groups


Author: Derek Robinson
Publisher: Springer Science & Business Media
ISBN: 1468401289
Category: Mathematics
Page: 481
View: 6109
" A group is defined by means of the laws of combinations of its symbols," according to a celebrated dictum of Cayley. And this is probably still as good a one-line explanation as any. The concept of a group is surely one of the central ideas of mathematics. Certainly there are a few branches of that science in which groups are not employed implicitly or explicitly. Nor is the use of groups confined to pure mathematics. Quantum theory, molecular and atomic structure, and crystallography are just a few of the areas of science in which the idea of a group as a measure of symmetry has played an important part. The theory of groups is the oldest branch of modern algebra. Its origins are to be found in the work of Joseph Louis Lagrange (1736-1813), Paulo Ruffini (1765-1822), and Evariste Galois (1811-1832) on the theory of algebraic equations. Their groups consisted of permutations of the variables or of the roots of polynomials, and indeed for much of the nineteenth century all groups were finite permutation groups. Nevertheless many of the fundamental ideas of group theory were introduced by these early workers and their successors, Augustin Louis Cauchy (1789-1857), Ludwig Sylow (1832-1918), Camille Jordan (1838-1922) among others. The concept of an abstract group is clearly recognizable in the work of Arthur Cayley (1821-1895) but it did not really win widespread acceptance until Walther von Dyck (1856-1934) introduced presentations of groups.

    • Mathematics

Permutation Groups


Author: John D. Dixon,Brian Mortimer
Publisher: Springer Science & Business Media
ISBN: 1461207312
Category: Mathematics
Page: 348
View: 9264
Following the basic ideas, standard constructions and important examples in the theory of permutation groups, the book goes on to develop the combinatorial and group theoretic structure of primitive groups leading to the proof of the pivotal ONan-Scott Theorem which links finite primitive groups with finite simple groups. Special topics covered include the Mathieu groups, multiply transitive groups, and recent work on the subgroups of the infinite symmetric groups. With its many exercises and detailed references to the current literature, this text can serve as an introduction to permutation groups in a course at the graduate or advanced undergraduate level, as well as for self-study.

    • Mathematics

Representations of Algebraic Groups, Quantum Groups and Lie Algebras

AMS-IMS-SIAM Joint Summer Research Conference, July 11-15, 2004, Snowbird Resort, Snowbird, Utah
Author: Georgia Benkart
Publisher: American Mathematical Soc.
ISBN: 0821839241
Category: Mathematics
Page: 254
View: 815
The book contains several well-written, accessible survey papers in many interrelated areas of current research. These areas cover various aspects of the representation theory of Lie algebras, finite groups of Lie types, Hecke algebras, and Lie super algebras. Geometric methods have been instrumental in representation theory, and these proceedings include surveys on geometric as well as combinatorial constructions of the crystal basis for representations of quantum groups. Humphreys' paper outlines intricate connections among irreducible representations of certain blocks of reduced enveloping algebras of semi-simple Lie algebras in positive characteristic, left cells in two sided cells of affine Weyl groups, and the geometry of the nilpotent orbits. All these papers provide the reader with a broad picture of the interaction of many different research areas and should be helpful to those who want to have a glimpse of current research involving representation theory.

    • Mathematics

Quantum Groups, Quantum Categories and Quantum Field Theory


Author: Jürg Fröhlich,Thomas Kerler
Publisher: Springer
ISBN: 3540476113
Category: Mathematics
Page: 432
View: 8581
This book reviews recent results on low-dimensional quantum field theories and their connection with quantum group theory and the theory of braided, balanced tensor categories. It presents detailed, mathematically precise introductions to these subjects and then continues with new results. Among the main results are a detailed analysis of the representation theory of U (sl ), for q a primitive root of unity, and a semi-simple quotient thereof, a classfication of braided tensor categories generated by an object of q-dimension less than two, and an application of these results to the theory of sectors in algebraic quantum field theory. This clarifies the notion of "quantized symmetries" in quantum fieldtheory. The reader is expected to be familiar with basic notions and resultsin algebra. The book is intended for research mathematicians, mathematical physicists and graduate students.

    • Mathematics

Lie Groups, Lie Algebras, and Representations

An Elementary Introduction
Author: Brian Hall
Publisher: Springer Science & Business Media
ISBN: 9780387401225
Category: Mathematics
Page: 351
View: 6468
Lie groups, Lie algebras, and representation theory are the main focus of this text. In order to keep the prerequisites to a minimum, the author restricts attention to matrix Lie groups and Lie algebras. This approach keeps the discussion concrete, allows the reader to get to the heart of the subject quickly, and covers all of the most interesting examples. The book also introduces the often-intimidating machinery of roots and the Weyl group in a gradual way, using examples and representation theory as motivation. The text is divided into two parts. The first covers Lie groups and Lie algebras and the relationship between them, along with basic representation theory. The second part covers the theory of semisimple Lie groups and Lie algebras, beginning with a detailed analysis of the representations of SU(3). The author illustrates the general theory with numerous images pertaining to Lie algebras of rank two and rank three, including images of root systems, lattices of dominant integral weights, and weight diagrams. This book is sure to become a standard textbook for graduate students in mathematics and physics with little or no prior exposure to Lie theory. Brian Hall is an Associate Professor of Mathematics at the University of Notre Dame.

    • Science

Quantum Groups and Their Representations


Author: Anatoli Klimyk,Konrad Schmüdgen
Publisher: Springer Science & Business Media
ISBN: 3642608965
Category: Science
Page: 552
View: 335

    • Mathematics

Quantum Groups

A Path to Current Algebra
Author: Ross Street
Publisher: Cambridge University Press
ISBN: 1139461443
Category: Mathematics
Page: N.A
View: 2943
Algebra has moved well beyond the topics discussed in standard undergraduate texts on 'modern algebra'. Those books typically dealt with algebraic structures such as groups, rings and fields: still very important concepts! However Quantum Groups: A Path to Current Algebra is written for the reader at ease with at least one such structure and keen to learn algebraic concepts and techniques. A key to understanding these new developments is categorical duality. A quantum group is a vector space with structure. Part of the structure is standard: a multiplication making it an 'algebra'. Another part is not in those standard books at all: a comultiplication, which is dual to multiplication in the precise sense of category theory, making it a 'coalgebra'. While coalgebras, bialgebras and Hopf algebras have been around for half a century, the term 'quantum group', along with revolutionary new examples, was launched by Drinfel'd in 1986.

    • Mathematics

Lectures on Quantum Groups


Author: Jens Carsten Jantzen
Publisher: American Mathematical Soc.
ISBN: 9780821872345
Category: Mathematics
Page: 266
View: 3934
Starting with the quantum analog of sl2, the author carefully leads the reader through all the details necessary for full understanding of the subject, particularly emphasizing similarities and differences with the classical theory. The final chapters of the book describe the Kashiwara-Lusztig theory of so-called crystal (or canonical) bases in representations of complex semisimple Lie algebra.

    • Mathematics

Recent Advances in Representation Theory, Quantum Groups, Algebraic Geometry, and Related Topics


Author: Pramod M. Achar,Dijana Jakelić,Kailash C. Misra,Milen Yakimov
Publisher: American Mathematical Society
ISBN: 0821898523
Category: Mathematics
Page: 280
View: 1020
This volume contains the proceedings of two AMS Special Sessions "Geometric and Algebraic Aspects of Representation Theory" and "Quantum Groups and Noncommutative Algebraic Geometry" held October 13–14, 2012, at Tulane University, New Orleans, Louisiana. Included in this volume are original research and some survey articles on various aspects of representations of algebras including Kac—Moody algebras, Lie superalgebras, quantum groups, toroidal algebras, Leibniz algebras and their connections with other areas of mathematics and mathematical physics.