Download or read book Quantum Groups Quantum Categories and Quantum Field Theory written by Jürg Fröhlich and published by Springer. This book was released on 2006-11-15 with total page 438 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Quantum Groups Quantum Categories and Quantum Field Theory written by Jurg Frohlich and published by . This book was released on 2014-09-01 with total page 440 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Quantum Groups written by Christian Kassel and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 540 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Introduction to Quantum Groups written by Masud Chaichian and published by World Scientific. This book was released on 1996 with total page 362 pages. Available in PDF, EPUB and Kindle. Book excerpt: In the past decade there has been an extemely rapid growth in the interest and development of quantum group theory.This book provides students and researchers with a practical introduction to the principal ideas of quantum groups theory and its applications to quantum mechanical and modern field theory problems. It begins with a review of, and introduction to, the mathematical aspects of quantum deformation of classical groups, Lie algebras and related objects (algebras of functions on spaces, differential and integral calculi). In the subsequent chapters the richness of mathematical structure and power of the quantum deformation methods and non-commutative geometry is illustrated on the different examples starting from the simplest quantum mechanical system — harmonic oscillator and ending with actual problems of modern field theory, such as the attempts to construct lattice-like regularization consistent with space-time Poincaré symmetry and to incorporate Higgs fields in the general geometrical frame of gauge theories. Graduate students and researchers studying the problems of quantum field theory, particle physics and mathematical aspects of quantum symmetries will find the book of interest.
Download or read book Introduction to Quantum Group and Integrable Massive Models of Quantum Field Theory written by Mo-Lin Ge and published by World Scientific. This book was released on 1990-09-24 with total page 208 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Proceedings consists of 6 lectures each from Prof L Takhtajan and Prof F Smirnov which were presented during the workshop. Contents:Lectures on Integrable Massive Models of Quantum Field Theory (F A Smirnov):General Problems of the Quantum Field Theory. Completely Integrable ModelsSpace of States. Form Factors. A Set of Axioms for Form FactorsLocal Commutativity and Asymptotic ConditionsForm Factors in SU(2)-Invariant Thirring ModelNecessary Properties of the Currents Form Factors in SU(2)-Invariant Thirring ModelProperties of Currents in SU(2)-Invariant Thirring ModelLectures on Quantum Groups (L A Takhtajan):Historical Introduction. Algebraic BackgroundPoisson-Lie Groups, CYBE and Modified CYBE. Connection with ISM. Lie-Algebraic Meaning of CYBE and Modified CYBEQuantization Procedure as a Deformation of the Algebra of Classical ObservablesWeyl Quantization. Quantization of Poisson-Lie Groups Associated with CYBEQYBEQuantization of Poisson-Lie Groups Associated with Modified CYBE. Quantum Matrix Algebras. Quantum Determinant and Quantum Groups SL(n) and GL (n)Quantum Vector Spaces for the Quantum Groups SLq(n), GLq(n) and their Real Forms. Quantum Groups Oq(N), SPq(n), Quantum Vector Spaces Associated with them and their Real FormsQuantization of the Universal Enveloping Algebras of the Simple Lie Algebras. Elements of the Representations Theory Readership: Mathematical physicists. keywords:
Download or read book Foundations of Quantum Group Theory written by Shahn Majid and published by Cambridge University Press. This book was released on 2000 with total page 668 pages. Available in PDF, EPUB and Kindle. Book excerpt: A graduate level text which systematically lays out the foundations of Quantum Groups.
Download or read book Quantum Invariants of Knots and 3 Manifolds written by Vladimir G. Turaev and published by Walter de Gruyter GmbH & Co KG. This book was released on 2020-03-23 with total page 600 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph provides a systematic treatment of topological quantum field theories (TQFT's) in three dimensions, inspired by the discovery of the Jones polynomial of knots, the Witten-Chern-Simons field theory, and the theory of quantum groups. The author, one of the leading experts in the subject, gives a rigorous and self-contained exposition of new fundamental algebraic and topological concepts that emerged in this theory. The book is divided into three parts. Part I presents a construction of 3-dimensional TQFT's and 2-dimensional modular functors from so-called modular categories. This gives new knot and 3-manifold invariants as well as linear representations of the mapping class groups of surfaces. In Part II the machinery of 6j-symbols is used to define state sum invariants of 3-manifolds. Their relation to the TQFT's constructed in Part I is established via the theory of shadows. Part III provides constructions of modular categories, based on quantum groups and Kauffman's skein modules. This book is accessible to graduate students in mathematics and physics with a knowledge of basic algebra and topology. It will be an indispensable source for everyone who wishes to enter the forefront of this rapidly growing and fascinating area at the borderline of mathematics and physics. Most of the results and techniques presented here appear in book form for the first time.
Download or read book Homotopy Quantum Field Theory written by Vladimir G. Turaev and published by European Mathematical Society. This book was released on 2010 with total page 300 pages. Available in PDF, EPUB and Kindle. Book excerpt: Homotopy Quantum Field Theory (HQFT) is a branch of Topological Quantum Field Theory founded by E. Witten and M. Atiyah. It applies ideas from theoretical physics to study principal bundles over manifolds and, more generally, homotopy classes of maps from manifolds to a fixed target space. This book is the first systematic exposition of Homotopy Quantum Field Theory. It starts with a formal definition of an HQFT and provides examples of HQFTs in all dimensions. The main body of the text is focused on $2$-dimensional and $3$-dimensional HQFTs. A study of these HQFTs leads to new algebraic objects: crossed Frobenius group-algebras, crossed ribbon group-categories, and Hopf group-coalgebras. These notions and their connections with HQFTs are discussed in detail. The text ends with several appendices including an outline of recent developments and a list of open problems. Three appendices by M. Muger and A. Virelizier summarize their work in this area. The book is addressed to mathematicians, theoretical physicists, and graduate students interested in topological aspects of quantum field theory. The exposition is self-contained and well suited for a one-semester graduate course. Prerequisites include only basics of algebra and topology.
Download or read book Quantum Topology written by Louis H Kauffman and published by World Scientific. This book was released on 1993-09-15 with total page 392 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes a review volume on the relatively new subject of Quantum Topology. Quantum Topology has its inception in the 1984/1985 discoveries of new invariants of knots and links (Jones, Homfly and Kauffman polynomials). These invariants were rapidly connected with quantum groups and methods in statistical mechanics. This was followed by Edward Witten's introduction of methods of quantum field theory into the subject and the formulation by Witten and Michael Atiyah of the concept of topological quantum field theories. This book is a review volume of on-going research activity. The papers derive from talks given at the Special Session on Knot and Topological Quantum Field Theory of the American Mathematical Society held at Dayton, Ohio in the fall of 1992. The book consists of a self-contained article by Kauffman, entitled Introduction to Quantum Topology and eighteen research articles by participants in the special session. This book should provide a useful source of ideas and results for anyone interested in the interface between topology and quantum field theory. Contents:Introduction to Quantum Topology (L H Kauffman)Knot Theory, Exotic Spheres and Global Gravitational Anomalies (R A Baadhio)A Diagrammatic Theory of Knotted Surfaces (J S Carter & M Saito)A Categorical Construction of 4D Topological Quantum Field Theories (L Crane & D Yetter)Evaluating the Crane-Yetter Invariant (L Crane, L H Kauffman & D Yetter)A Method for Computing the Arf Invariants of Links (P Gilmer)Triangulations, Categories and Extended Topological Field Theories (R J Lawrence)The Casson Invariant for Two-Fold Branched Covers of Links (D Mullins)Elementary Conjectures in Classical Knot Theory (J H Przytycki)Knot Polynomials as States of Nonperturbative Four Dimensional Quantum Gravity (J Pullin)On Invariants of 3-Manifolds Derived from Abelian Groups (J Mattes, M M Polyak & N Reshetikhin)and other papers Readership: Mathematicians and mathematical physicists. keywords:Quantum Topology;Topological Quantum Field Theory;Meeting;AMS Special Session;Dayton, OH (USA)
Download or read book Quantum Invariants of Knots and 3 Manifolds written by Vladimir G. Turaev and published by Walter de Gruyter GmbH & Co KG. This book was released on 2016-07-11 with total page 608 pages. Available in PDF, EPUB and Kindle. Book excerpt: Due to the strong appeal and wide use of this monograph, it is now available in its third revised edition. The monograph gives a systematic treatment of 3-dimensional topological quantum field theories (TQFTs) based on the work of the author with N. Reshetikhin and O. Viro. This subject was inspired by the discovery of the Jones polynomial of knots and the Witten-Chern-Simons field theory. On the algebraic side, the study of 3-dimensional TQFTs has been influenced by the theory of braided categories and the theory of quantum groups.The book is divided into three parts. Part I presents a construction of 3-dimensional TQFTs and 2-dimensional modular functors from so-called modular categories. This gives a vast class of knot invariants and 3-manifold invariants as well as a class of linear representations of the mapping class groups of surfaces. In Part II the technique of 6j-symbols is used to define state sum invariants of 3-manifolds. Their relation to the TQFTs constructed in Part I is established via the theory of shadows. Part III provides constructions of modular categories, based on quantum groups and skein modules of tangles in the 3-space.This fundamental contribution to topological quantum field theory is accessible to graduate students in mathematics and physics with knowledge of basic algebra and topology. It is an indispensable source for everyone who wishes to enter the forefront of this fascinating area at the borderline of mathematics and physics. Contents:Invariants of graphs in Euclidean 3-space and of closed 3-manifoldsFoundations of topological quantum field theoryThree-dimensional topological quantum field theoryTwo-dimensional modular functors6j-symbolsSimplicial state sums on 3-manifoldsShadows of manifolds and state sums on shadowsConstructions of modular categories
Download or read book Quantum Groups and Noncommutative Geometry written by Yuri I. Manin and published by Springer. This book was released on 2018-10-11 with total page 125 pages. Available in PDF, EPUB and Kindle. Book excerpt: This textbook presents the second edition of Manin's celebrated 1988 Montreal lectures, which influenced a new generation of researchers in algebra to take up the study of Hopf algebras and quantum groups. In this expanded write-up of those lectures, Manin systematically develops an approach to quantum groups as symmetry objects in noncommutative geometry in contrast to the more deformation-oriented approach due to Faddeev, Drinfeld, and others. This new edition contains an extra chapter by Theo Raedschelders and Michel Van den Bergh, surveying recent work that focuses on the representation theory of a number of bi- and Hopf algebras that were first introduced in Manin's lectures, and have since gained a lot of attention. Emphasis is placed on the Tannaka–Krein formalism, which further strengthens Manin's approach to symmetry and moduli-objects in noncommutative geometry.
Download or read book A Quantum Groups Primer written by Shahn Majid and published by Cambridge University Press. This book was released on 2002-04-04 with total page 183 pages. Available in PDF, EPUB and Kindle. Book excerpt: Self-contained introduction to quantum groups as algebraic objects, suitable as a textbook for graduate courses.
Download or read book Categories for Quantum Theory written by Chris Heunen and published by Oxford University Press. This book was released on 2019-11-14 with total page 320 pages. Available in PDF, EPUB and Kindle. Book excerpt: Monoidal category theory serves as a powerful framework for describing logical aspects of quantum theory, giving an abstract language for parallel and sequential composition, and a conceptual way to understand many high-level quantum phenomena. This text lays the foundation for this categorical quantum mechanics, with an emphasis on the graphical calculus which makes computation intuitive. Biproducts and dual objects are introduced and used to model superposition and entanglement, with quantum teleportation studied abstractly using these structures. Monoids, Frobenius structures and Hopf algebras are described, and it is shown how they can be used to model classical information and complementary observables. The CP construction, a categorical tool to describe probabilistic quantum systems, is also investigated. The last chapter introduces higher categories, surface diagrams and 2-Hilbert spaces, and shows how the language of duality in monoidal 2-categories can be used to reason about quantum protocols, including quantum teleportation and dense coding. Prior knowledge of linear algebra, quantum information or category theory would give an ideal background for studying this text, but it is not assumed, with essential background material given in a self-contained introductory chapter. Throughout the text links with many other areas are highlighted, such as representation theory, topology, quantum algebra, knot theory, and probability theory, and nonstandard models are presented, such as sets and relations. All results are stated rigorously, and full proofs are given as far as possible, making this book an invaluable reference for modern techniques in quantum logic, with much of the material not available in any other textbook.
Download or read book Quantum Group Symmetry and Q tensor Algebras written by L. C. Biedenharn and published by World Scientific. This book was released on 1995 with total page 305 pages. Available in PDF, EPUB and Kindle. Book excerpt: Quantum groups are a generalization of the classical Lie groups and Lie algebras and provide a natural extension of the concept of symmetry fundamental to physics. This monograph is a survey of the major developments in quantum groups, using an original approach based on the fundamental concept of a tensor operator. Using this concept, properties of both the algebra and co-algebra are developed from a single uniform point of view, which is especially helpful for understanding the noncommuting co-ordinates of the quantum plane, which we interpret as elementary tensor operators. Representations of the q-deformed angular momentum group are discussed, including the case where q is a root of unity, and general results are obtained for all unitary quantum groups using the method of algebraic induction. Tensor operators are defined and discussed with examples, and a systematic treatment of the important (3j) series of operators is developed in detail. This book is a good reference for graduate students in physics and mathematics.
Download or read book Introduction to the Quantum Yang Baxter Equation and Quantum Groups An Algebraic Approach written by L.A. Lambe and published by Springer Science & Business Media. This book was released on 2013-11-22 with total page 314 pages. Available in PDF, EPUB and Kindle. Book excerpt: Chapter 1 The algebraic prerequisites for the book are covered here and in the appendix. This chapter should be used as reference material and should be consulted as needed. A systematic treatment of algebras, coalgebras, bialgebras, Hopf algebras, and represen tations of these objects to the extent needed for the book is given. The material here not specifically cited can be found for the most part in [Sweedler, 1969] in one form or another, with a few exceptions. A great deal of emphasis is placed on the coalgebra which is the dual of n x n matrices over a field. This is the most basic example of a coalgebra for our purposes and is at the heart of most algebraic constructions described in this book. We have found pointed bialgebras useful in connection with solving the quantum Yang-Baxter equation. For this reason we develop their theory in some detail. The class of examples described in Chapter 6 in connection with the quantum double consists of pointed Hopf algebras. We note the quantized enveloping algebras described Hopf algebras. Thus for many reasons pointed bialgebras are elsewhere are pointed of fundamental interest in the study of the quantum Yang-Baxter equation and objects quantum groups.
Download or read book Quantum Group And Quantum Integrable Systems Nankai Lectures On Mathematical Physics written by Mo-lin Ge and published by World Scientific. This book was released on 1992-05-30 with total page 242 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains the lectures given by the three speakers, M Jimbo, P P Kulish and E K Sklyanin, who are outstanding experts in their field. It is essential reading to those working in the fields of Quantum Groups, and Integrable Systems.
Download or read book Factorization Algebras in Quantum Field Theory written by Kevin Costello and published by Cambridge University Press. This book was released on 2017 with total page 399 pages. Available in PDF, EPUB and Kindle. Book excerpt: This first volume develops factorization algebras with a focus upon examples exhibiting their use in field theory, which will be useful for researchers and graduates.
