Download or read book An Invitation to Quantum Groups and Duality written by Thomas Timmermann and published by European Mathematical Society. This book was released on 2008 with total page 436 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides an introduction to the theory of quantum groups with emphasis on their duality and on the setting of operator algebras. Part I of the text presents the basic theory of Hopf algebras, Van Daele's duality theory of algebraic quantum groups, and Woronowicz's compact quantum groups, staying in a purely algebraic setting. Part II focuses on quantum groups in the setting of operator algebras. Woronowicz's compact quantum groups are treated in the setting of $C^*$-algebras, and the fundamental multiplicative unitaries of Baaj and Skandalis are studied in detail. An outline of Kustermans' and Vaes' comprehensive theory of locally compact quantum groups completes this part. Part III leads to selected topics, such as coactions, Baaj-Skandalis-duality, and approaches to quantum groupoids in 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, $C^*$-algebras, and von Neumann algebras.
Download or read book Quantum Isometry Groups written by Debashish Goswami and published by Springer. This book was released on 2017-01-05 with total page 254 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Compact Matrix Quantum Groups and Their Combinatorics written by Amaury Freslon and published by Cambridge University Press. This book was released on 2023-07-27 with total page 302 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Quantum Field Theory III Gauge Theory written by Eberhard Zeidler and published by Springer Science & Business Media. This book was released on 2011-08-17 with total page 1141 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this third volume of his modern introduction to quantum field theory, Eberhard Zeidler examines the mathematical and physical aspects of gauge theory as a principle tool for describing the four fundamental forces which act in the universe: gravitative, electromagnetic, weak interaction and strong interaction. Volume III concentrates on the classical aspects of gauge theory, describing the four fundamental forces by the curvature of appropriate fiber bundles. This must be supplemented by the crucial, but elusive quantization procedure. The book is arranged in four sections, devoted to realizing the universal principle force equals curvature: Part I: The Euclidean Manifold as a Paradigm Part II: Ariadne's Thread in Gauge Theory Part III: Einstein's Theory of Special Relativity Part IV: Ariadne's Thread in Cohomology For students of mathematics the book is designed to demonstrate that detailed knowledge of the physical background helps to reveal interesting interrelationships among diverse mathematical topics. Physics students will be exposed to a fairly advanced mathematics, beyond the level covered in the typical physics curriculum. Quantum Field Theory builds a bridge between mathematicians and physicists, based on challenging questions about the fundamental forces in the universe (macrocosmos), and in the world of elementary particles (microcosmos).
Download or read book Analysis and Quantum Groups written by Lars Tuset and published by Springer Nature. This book was released on 2022-07-27 with total page 632 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume presents a completely self-contained introduction to the elaborate theory of locally compact quantum groups, bringing the reader to the frontiers of present-day research. The exposition includes a substantial amount of material on functional analysis and operator algebras, subjects which in themselves have become increasingly important with the advent of quantum information theory. In particular, the rather unfamiliar modular theory of weights plays a crucial role in the theory, due to the presence of ‘Haar integrals’ on locally compact quantum groups, and is thus treated quite extensively The topics covered are developed independently, and each can serve either as a separate course in its own right or as part of a broader course on locally compact quantum groups. The second part of the book covers crossed products of coactions, their relation to subfactors and other types of natural products such as cocycle bicrossed products, quantum doubles and doublecrossed products. Induced corepresentations, Galois objects and deformations of coactions by cocycles are also treated. Each section is followed by a generous supply of exercises. To complete the book, an appendix is provided on topology, measure theory and complex function theory.
Download or read book Topological Phases of Matter and Quantum Computation written by Paul Bruillard and published by American Mathematical Soc.. This book was released on 2020-03-31 with total page 242 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains the proceedings of the AMS Special Session on Topological Phases of Matter and Quantum Computation, held from September 24–25, 2016, at Bowdoin College, Brunswick, Maine. Topological quantum computing has exploded in popularity in recent years. Sitting at the triple point between mathematics, physics, and computer science, it has the potential to revolutionize sub-disciplines in these fields. The academic importance of this field has been recognized in physics through the 2016 Nobel Prize. In mathematics, some of the 1990 Fields Medals were awarded for developments in topics that nowadays are fundamental tools for the study of topological quantum computation. Moreover, the practical importance of this discipline has been underscored by recent industry investments. The relative youth of this field combined with a high degree of interest in it makes now an excellent time to get involved. Furthermore, the cross-disciplinary nature of topological quantum computing provides an unprecedented number of opportunities for cross-pollination of mathematics, physics, and computer science. This can be seen in the variety of works contained in this volume. With articles coming from mathematics, physics, and computer science, this volume aims to provide a taste of different sub-disciplines for novices and a wealth of new perspectives for veteran researchers. Regardless of your point of entry into topological quantum computing or your experience level, this volume has something for you.
Download or read book Algebraic Methods in Functional Analysis written by Ivan G. Todorov and published by Springer Science & Business Media. This book was released on 2013-10-25 with total page 301 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume comprises the proceedings of the Conference on Operator Theory and its Applications held in Gothenburg, Sweden, April 26-29, 2011. The conference was held in honour of Professor Victor Shulman on the occasion of his 65th birthday. The papers included in the volume cover a large variety of topics, among them the theory of operator ideals, linear preservers, C*-algebras, invariant subspaces, non-commutative harmonic analysis, and quantum groups, and reflect recent developments in these areas. The book consists of both original research papers and high quality survey articles, all of which were carefully refereed.
Download or read book Advances in Functional Analysis and Operator Theory written by Marat V. Markin and published by American Mathematical Society. This book was released on 2024-04-09 with total page 250 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains the proceedings of the AMS-EMS-SMF Special Session on Advances in Functional Analysis and Operator Theory, held July 18–22, 2022, at the Université de Grenoble-Alpes, Grenoble, France. The papers reflect the modern interplay between differential equations, functional analysis, operator algebras, and their applications from the dynamics to quantum groups to number theory. Among the topics discussed are the Sturm-Liouville and boundary value problems, axioms of quantum mechanics, $C^{*}$-algebras and symbolic dynamics, von Neumann algebras and low-dimensional topology, quantum permutation groups, the Jordan algebras, and the Kadison–Singer transforms.
Download or read book Algebras Rings and Modules written by Michiel Hazewinkel and published by American Mathematical Soc.. This book was released on 2010 with total page 425 pages. Available in PDF, EPUB and Kindle. Book excerpt: Presenting an introduction to the theory of Hopf algebras, the authors also discuss some important aspects of the theory of Lie algebras. This book includes a chapters on the Hopf algebra of symmetric functions, the Hopf algebra of representations of the symmetric groups, the Hopf algebras of the nonsymmetric and quasisymmetric functions, and the Hopf algebra of permutations.
Download or read book Noncommutative Mathematics for Quantum Systems written by Uwe Franz and published by Cambridge University Press. This book was released on 2016-01-07 with total page 200 pages. Available in PDF, EPUB and Kindle. Book excerpt: Noncommutative mathematics is a significant new trend of mathematics. Initially motivated by the development of quantum physics, the idea of 'making theory noncommutative' has been extended to many areas of pure and applied mathematics. This book is divided into two parts. The first part provides an introduction to quantum probability, focusing on the notion of independence in quantum probability and on the theory of quantum stochastic processes with independent and stationary increments. The second part provides an introduction to quantum dynamical systems, discussing analogies with fundamental problems studied in classical dynamics. The desire to build an extension of the classical theory provides new, original ways to understand well-known 'commutative' results. On the other hand the richness of the quantum mathematical world presents completely novel phenomena, never encountered in the classical setting. This book will be useful to students and researchers in noncommutative probability, mathematical physics and operator algebras.
Download or read book An Invitation to Noncommutative Geometry written by Masoud Khalkhali and published by World Scientific. This book was released on 2008 with total page 515 pages. Available in PDF, EPUB and Kindle. Book excerpt: A walk in the noncommutative garden / A. Connes and M. Marcolli -- Renormalization of noncommutative quantum field theory / H. Grosse and R. Wulkenhaar -- Lectures on noncommutative geometry / M. Khalkhali -- Noncommutative bundles and instantons in Tehran / G. Landi and W. D. van Suijlekom -- Lecture notes on noncommutative algebraic geometry and noncommutative tori / S. Mahanta -- Lectures on derived and triangulated categories / B. Noohi -- Examples of noncommutative manifolds: complex tori and spherical manifolds / J. Plazas -- D-branes in noncommutative field theory / R. J. Szabo.
Download or read book Noncommutative Geometry written by Igor V. Nikolaev and published by Walter de Gruyter GmbH & Co KG. This book was released on 2022-07-18 with total page 400 pages. Available in PDF, EPUB and Kindle. Book excerpt: Noncommutative geometry studies an interplay between spatial forms and algebras with non-commutative multiplication. This book covers the key concepts of noncommutative geometry and its applications in topology, algebraic geometry, and number theory. Our presentation is accessible to the graduate students as well as nonexperts in the field. The second edition includes two new chapters on arithmetic topology and quantum arithmetic.
Download or read book Introduction to Group Theory written by Oleg Vladimirovič Bogopolʹskij and published by European Mathematical Society. This book was released on 2008 with total page 196 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book quickly introduces beginners to general group theory and then focuses on three main themes : finite group theory, including sporadic groups combinatorial and geometric group theory, including the Bass-Serre theory of groups acting on trees the theory of train tracks by Bestvina and Handel for automorphisms of free groups With its many examples, exercises, and full solutions to selected exercises, this text provides a gentle introduction that is ideal for self-study and an excellent preparation for applications. A distinguished feature of the presentation is that algebraic and geometric techniques are balanced. The beautiful theory of train tracks is illustrated by two nontrivial examples. Presupposing only a basic knowledge of algebra, the book is addressed to anyone interested in group theory: from advanced undergraduate and graduate students to specialists.
Download or read book Model Theory of Operator Algebras written by Isaac Goldbring and published by Walter de Gruyter GmbH & Co KG. This book was released on 2023-07-24 with total page 651 pages. Available in PDF, EPUB and Kindle. Book excerpt: Continuous model theory is an extension of classical first order logic which is best suited for classes of structures which are endowed with a metric. Applications have grown considerably in the past decade. This book is dedicated to showing how the techniques of continuous model theory are used to study C*-algebras and von Neumann algebras. This book geared to researchers in both logic and functional analysis provides the first self-contained collection of articles surveying the many applications of continuous logic to operator algebras that have been obtained in the last 15 years.
Download or read book Feynman s Operational Calculus and Beyond written by Gerald W. Johnson and published by Oxford Mathematical Monographs. This book was released on 2015 with total page 385 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is aimed at providing a coherent, essentially self-contained, rigorous and comprehensive abstract theory of Feynman's operational calculus for noncommuting operators. Although it is inspired by Feynman's original heuristic suggestions and time-ordering rules in his seminal 1951 paper An operator calculus having applications in quantum electrodynamics, as will be made abundantly clear in the introduction (Chapter 1) and elsewhere in the text, the theory developed in this book also goes well beyond them in a number of directions which were not anticipated in Feynman's work. Hence, the second part of the main title of this book. The basic properties of the operational calculus are developed and certain algebraic and analytic properties of the operational calculus are explored. Also, the operational calculus will be seen to possess some pleasant stability properties. Furthermore, an evolution equation and a generalized integral equation obeyed by the operational calculus are discussed and connections with certain analytic Feynman integrals are noted. This volume is essentially self-contained and we only assume that the reader has a reasonable, graduate level, background in analysis, measure theory and functional analysis or operator theory. Much of the necessary remaining background is supplied in the text itself.
Download or read book Fusion of Defects written by Arthur Bartels and published by American Mathematical Soc.. This book was released on 2019-04-10 with total page 114 pages. Available in PDF, EPUB and Kindle. Book excerpt: Conformal nets provide a mathematical model for conformal field theory. The authors define a notion of defect between conformal nets, formalizing the idea of an interaction between two conformal field theories. They introduce an operation of fusion of defects, and prove that the fusion of two defects is again a defect, provided the fusion occurs over a conformal net of finite index. There is a notion of sector (or bimodule) between two defects, and operations of horizontal and vertical fusion of such sectors. The authors' most difficult technical result is that the horizontal fusion of the vacuum sectors of two defects is isomorphic to the vacuum sector of the fused defect. Equipped with this isomorphism, they construct the basic interchange isomorphism between the horizontal fusion of two vertical fusions and the vertical fusion of two horizontal fusions of sectors.
Download or read book Crossed Products of C Algebras Topological Dynamics and Classification written by Thierry Giordano and published by Springer. This book was released on 2018-08-28 with total page 494 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book collects the notes of the lectures given at an Advanced Course on Dynamical Systems at the Centre de Recerca Matemàtica (CRM) in Barcelona. The notes consist of four series of lectures. The first one, given by Andrew Toms, presents the basic properties of the Cuntz semigroup and its role in the classification program of simple, nuclear, separable C*-algebras. The second series of lectures, delivered by N. Christopher Phillips, serves as an introduction to group actions on C*-algebras and their crossed products, with emphasis on the simple case and when the crossed products are classifiable. The third one, given by David Kerr, treats various developments related to measure-theoretic and topological aspects of crossed products, focusing on internal and external approximation concepts, both for groups and C*-algebras. Finally, the last series of lectures, delivered by Thierry Giordano, is devoted to the theory of topological orbit equivalence, with particular attention to the classification of minimal actions by finitely generated abelian groups on the Cantor set.
