Compact Matrix Quantum Groups and Their Combinatorics PDF Download

Author: Amaury Freslon
Publisher: Cambridge University Press
ISBN: 1009345680
Category : Mathematics
Languages : en
Pages : 302

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Author: Amaury Freslon
Publisher: Cambridge University Press
ISBN: 9781009345699
Category : Mathematics
Languages : en
Pages : 0

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Author: Debashish Goswami
Publisher: Springer
ISBN: 813223667X
Category : Mathematics
Languages : en
Pages : 254

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This book offers an up-to-date overview of the recently proposed theory of quantum isometry groups. Written by the founders, it is the first book to present the research on the “quantum isometry group”, highlighting the interaction of noncommutative geometry and quantum groups, which is a noncommutative generalization of the notion of group of isometry of a classical Riemannian manifold. The motivation for this generalization is the importance of isometry groups in both mathematics and physics. The framework consists of Alain Connes’ “noncommutative geometry” and the operator-algebraic theory of “quantum groups”. The authors prove the existence of quantum isometry group for noncommutative manifolds given by spectral triples under mild conditions and discuss a number of methods for computing them. One of the most striking and profound findings is the non-existence of non-classical quantum isometry groups for arbitrary classical connected compact manifolds and, by using this, the authors explicitly describe quantum isometry groups of most of the noncommutative manifolds studied in the literature. Some physical motivations and possible applications are also discussed.

Author: Pavel Etingof
Publisher: American Mathematical Soc.
ISBN: 1470434415
Category : Mathematics
Languages : en
Pages : 362

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Is there a vector space whose dimension is the golden ratio? Of course not—the golden ratio is not an integer! But this can happen for generalizations of vector spaces—objects of a tensor category. The theory of tensor categories is a relatively new field of mathematics that generalizes the theory of group representations. It has deep connections with many other fields, including representation theory, Hopf algebras, operator algebras, low-dimensional topology (in particular, knot theory), homotopy theory, quantum mechanics and field theory, quantum computation, theory of motives, etc. This book gives a systematic introduction to this theory and a review of its applications. While giving a detailed overview of general tensor categories, it focuses especially on the theory of finite tensor categories and fusion categories (in particular, braided and modular ones), and discusses the main results about them with proofs. In particular, it shows how the main properties of finite-dimensional Hopf algebras may be derived from the theory of tensor categories. Many important results are presented as a sequence of exercises, which makes the book valuable for students and suitable for graduate courses. Many applications, connections to other areas, additional results, and references are discussed at the end of each chapter.

Author: V.A. Malyshev
Publisher: Springer Science & Business Media
ISBN: 9781402007927
Category : Science
Languages : en
Pages : 352

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New and striking results obtained in recent years from an intensive study of asymptotic combinatorics have led to a new, higher level of understanding of related problems: the theory of integrable systems, the Riemann-Hilbert problem, asymptotic representation theory, spectra of random matrices, combinatorics of Young diagrams and permutations, and even some aspects of quantum field theory.

Author: Robert Curtis
Publisher: Cambridge University Press
ISBN: 0521575877
Category : Mathematics
Languages : en
Pages : 315

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Book Description
Proceedings containing twenty articles by leading experts in group theory and its applications.

Author: Alexandru Nica
Publisher: Cambridge University Press
ISBN: 0521858526
Category : Mathematics
Languages : en
Pages : 430

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Book Description
This 2006 book is a self-contained introduction to free probability theory suitable for an introductory graduate level course.

Author: Jinho Baik
Publisher: American Mathematical Soc.
ISBN: 0821848410
Category : Mathematics
Languages : en
Pages : 478

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Over the last fifteen years a variety of problems in combinatorics have been solved in terms of random matrix theory. More precisely, the situation is as follows: the problems at hand are probabilistic in nature and, in an appropriate scaling limit, it turns out that certain key quantities associated with these problems behave statistically like the eigenvalues of a (large) random matrix. Said differently, random matrix theory provides a “stochastic special function theory” for a broad and growing class of problems in combinatorics. The goal of this book is to analyze in detail two key examples of this phenomenon, viz., Ulam's problem for increasing subsequences of random permutations and domino tilings of the Aztec diamond. Other examples are also described along the way, but in less detail. Techniques from many different areas in mathematics are needed to analyze these problems. These areas include combinatorics, probability theory, functional analysis, complex analysis, and the theory of integrable systems. The book is self-contained, and along the way we develop enough of the theory we need from each area that a general reader with, say, two or three years experience in graduate school can learn the subject directly from the text.

Author: Guillaume Aubrun
Publisher: Springer
ISBN: 331963206X
Category : Mathematics
Languages : en
Pages : 126

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Providing an introduction to current research topics in functional analysis and its applications to quantum physics, this book presents three lectures surveying recent progress and open problems. A special focus is given to the role of symmetry in non-commutative probability, in the theory of quantum groups, and in quantum physics. The first lecture presents the close connection between distributional symmetries and independence properties. The second introduces many structures (graphs, C*-algebras, discrete groups) whose quantum symmetries are much richer than their classical symmetry groups, and describes the associated quantum symmetry groups. The last lecture shows how functional analytic and geometric ideas can be used to detect and to quantify entanglement in high dimensions. The book will allow graduate students and young researchers to gain a better understanding of free probability, the theory of compact quantum groups, and applications of the theory of Banach spaces to quantum information. The latter applications will also be of interest to theoretical and mathematical physicists working in quantum theory.

Author: Richard P. Stanley
Publisher: Springer Science & Business Media
ISBN: 1461469988
Category : Mathematics
Languages : en
Pages : 226

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Book Description
Written by one of the foremost experts in the field, Algebraic Combinatorics is a unique undergraduate textbook that will prepare the next generation of pure and applied mathematicians. The combination of the author’s extensive knowledge of combinatorics and classical and practical tools from algebra will inspire motivated students to delve deeply into the fascinating interplay between algebra and combinatorics. Readers will be able to apply their newfound knowledge to mathematical, engineering, and business models. The text is primarily intended for use in a one-semester advanced undergraduate course in algebraic combinatorics, enumerative combinatorics, or graph theory. Prerequisites include a basic knowledge of linear algebra over a field, existence of finite fields, and group theory. The topics in each chapter build on one another and include extensive problem sets as well as hints to selected exercises. Key topics include walks on graphs, cubes and the Radon transform, the Matrix–Tree Theorem, and the Sperner property. There are also three appendices on purely enumerative aspects of combinatorics related to the chapter material: the RSK algorithm, plane partitions, and the enumeration of labeled trees. Richard Stanley is currently professor of Applied Mathematics at the Massachusetts Institute of Technology. Stanley has received several awards including the George Polya Prize in applied combinatorics, the Guggenheim Fellowship, and the Leroy P. Steele Prize for mathematical exposition. Also by the author: Combinatorics and Commutative Algebra, Second Edition, © Birkhauser.