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.
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.
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.
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.
Graduate lectures on the interface between mathematics and physics.
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.
Aus den Besprechungen: "Ein Mathematikbuch der Superlativen, für Mathematiker (jeder Schattierung) und Nichtmathematiker (denen völlig unbekannte Dimensionen der Mathematik eröffnet werden - künstlerische, magische, historische, philosophische, wissenschaftstheoretische, "unlogische", phantasieerfüllte usw.). Der Aufbau ist meisterhaft, die Lektüre höchst anregend und leicht lesbar." Monatshefte für Mathematik #1 "Ein gelungenes Werk, das dem Vorurteil entgegenwirkt, Mathematik bestehe nur aus isolierten Theorien." Die NEUE HOCHSCHULE #1 "Das Lesen ist ein Genuß, den man sich nicht entgehen lassen sollte." Jahresbericht der Deutschen Mathematiker-Vereinigung #1
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.
This book aims to present several new developments on stochastic processes and operator calculus on quantum groups. Topics which are treated include operator calculus, dual representations, stochastic processes and diffusions, Appell polynomials and systems in connection with evolution equations. Audience: This volume contains introductory material for graduate students who are new to the field, as well as more advanced material for specialists in probability theory, algebraic structures, representation theory, mathematical physics and theoretical physics.
The Encyclopedia of Mathematical Physics provides a complete resource for researchers, students and lecturers with an interest in mathematical physics. It enables readers to access basic information on topics peripheral to their own areas, to provide a repository of the core information in the area that can be used to refresh the researcher's own memory banks, and aid teachers in directing students to entries relevant to their course-work. The Encyclopedia does contain information that has been distilled, organised and presented as a complete reference tool to the user and a landmark to the body of knowledge that has accumulated in this domain. It also is a stimulus for new researchers working in mathematical physics or in areas using the methods originated from work in mathematical physics by providing them with focused high quality background information. * First comprehensive interdisciplinary coverage * Mathematical Physics explained to stimulate new developments and foster new applications of its methods to other fields * Written by an international group of experts * Contains several undergraduate-level introductory articles to facilitate acquisition of new expertise * Thematic index and extensive cross-referencing to provide easy access and quick search functionality * Also available online with active linking.

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