An Invitation To Quantum Groups And Duality

Author : Thomas Timmermann
Publisher : European Mathematical Society
Page : 436 pages
File Size : 44,6 Mb
Release : 2008
Category : Mathematics
ISBN : 3037190434

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An Invitation to Quantum Groups and Duality by Thomas Timmermann Pdf

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.

Author : Debashish Goswami,Jyotishman Bhowmick
Publisher : Springer
Page : 235 pages
File Size : 47,9 Mb
Release : 2017-01-05
Category : Mathematics
ISBN : 9788132236672

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Quantum Isometry Groups by Debashish Goswami,Jyotishman Bhowmick Pdf

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 : Amaury Freslon
Publisher : Cambridge University Press
Page : 302 pages
File Size : 42,9 Mb
Release : 2023-07-27
Category : Mathematics
ISBN : 9781009345682

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Compact Matrix Quantum Groups and Their Combinatorics by Amaury Freslon Pdf

Author : Eberhard Zeidler
Publisher : Springer Science & Business Media
Page : 1141 pages
File Size : 48,8 Mb
Release : 2011-08-17
Category : Mathematics
ISBN : 9783642224218

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Quantum Field Theory III: Gauge Theory by Eberhard Zeidler Pdf

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).

Author : Lars Tuset
Publisher : Springer Nature
Page : 632 pages
File Size : 43,5 Mb
Release : 2022-07-27
Category : Mathematics
ISBN : 9783031072468

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Analysis and Quantum Groups by Lars Tuset Pdf

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.

Author : Paul Bruillard,Carlos Ortiz Marrero,Julia Plavnik
Publisher : American Mathematical Soc.
Page : 240 pages
File Size : 52,9 Mb
Release : 2020-03-31
Category : Education
ISBN : 9781470440749

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Topological Phases of Matter and Quantum Computation by Paul Bruillard,Carlos Ortiz Marrero,Julia Plavnik Pdf

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.

Author : Ivan G. Todorov,Lyudmila Turowska
Publisher : Springer Science & Business Media
Page : 294 pages
File Size : 44,8 Mb
Release : 2013-10-25
Category : Mathematics
ISBN : 9783034805025

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Algebraic Methods in Functional Analysis by Ivan G. Todorov,Lyudmila Turowska Pdf

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. ​

Author : Marat V. Markin,Igor V. Nikolaev,Carsten Trunk
Publisher : American Mathematical Society
Page : 250 pages
File Size : 51,6 Mb
Release : 2024-04-09
Category : Mathematics
ISBN : 9781470473051

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Advances in Functional Analysis and Operator Theory by Marat V. Markin,Igor V. Nikolaev,Carsten Trunk Pdf

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.

Author : Michiel Hazewinkel,Nadezhda Mikhaĭlovna Gubareni,Vladimir V. Kirichenko
Publisher : American Mathematical Soc.
Page : 425 pages
File Size : 55,7 Mb
Release : 2010
Category : Mathematics
ISBN : 9780821852620

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Algebras, Rings and Modules by Michiel Hazewinkel,Nadezhda Mikhaĭlovna Gubareni,Vladimir V. Kirichenko Pdf

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.

Author : Igor V. Nikolaev
Publisher : Walter de Gruyter GmbH & Co KG
Page : 400 pages
File Size : 44,6 Mb
Release : 2022-07-18
Category : Mathematics
ISBN : 9783110788709

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Noncommutative Geometry by Igor V. Nikolaev Pdf

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.

Author : Oleg Vladimirovič Bogopolʹskij
Publisher : European Mathematical Society
Page : 196 pages
File Size : 44,9 Mb
Release : 2008
Category : Mathematics
ISBN : 3037190418

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Introduction to Group Theory by Oleg Vladimirovič Bogopolʹskij Pdf

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.

Author : Gerald W Johnson,Michel L. Lapidus,Lance Nielsen
Publisher : OUP Oxford
Page : 368 pages
File Size : 54,9 Mb
Release : 2015-08-06
Category : Science
ISBN : 9780191006876

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Feynman's Operational Calculus and Beyond by Gerald W Johnson,Michel L. Lapidus,Lance Nielsen Pdf

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.

Author : Matilde Marcolli,Masoud Khalkhali
Publisher : World Scientific
Page : 516 pages
File Size : 43,5 Mb
Release : 2008-02-11
Category : Science
ISBN : 9789814475624

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An Invitation To Noncommutative Geometry by Matilde Marcolli,Masoud Khalkhali Pdf

This is the first existing volume that collects lectures on this important and fast developing subject in mathematics. The lectures are given by leading experts in the field and the range of topics is kept as broad as possible by including both the algebraic and the differential aspects of noncommutative geometry as well as recent applications to theoretical physics and number theory.

Author : Sergio Duarte,Jean-Pierre Gazeau,Sofiane Faci,Tobias Micklitz,Ricardo Scherer,Francesco Toppan
Publisher : Springer
Page : 422 pages
File Size : 40,8 Mb
Release : 2018-01-09
Category : Science
ISBN : 9783319691640

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Physical and Mathematical Aspects of Symmetries by Sergio Duarte,Jean-Pierre Gazeau,Sofiane Faci,Tobias Micklitz,Ricardo Scherer,Francesco Toppan Pdf

This proceedings records the 31st International Colloquium on Group Theoretical Methods in Physics (“Group 31”). Plenary-invited articles propose new approaches to the moduli spaces in gauge theories (V. Pestun, 2016 Weyl Prize Awardee), the phenomenology of neutrinos in non-commutative space-time, the use of Hardy spaces in quantum physics, contradictions in the use of statistical methods on complex systems, and alternative models of supersymmetry. This volume’s survey articles broaden the colloquia’s scope out into Majorana neutrino behavior, the dynamics of radiating charges, statistical pattern recognition of amino acids, and a variety of applications of gauge theory, among others. This year’s proceedings further honors Bertram Kostant (2016 Wigner Medalist), as well as S.T. Ali and L. Boyle, for their life-long contributions to the math and physics communities. The aim of the ICGTMP is to provide a forum for physicists, mathematicians, and scientists of related disciplines who develop or apply methods in group theory to share their research. The 31st ICGTMP was held in Rio de Janeiro, Brazil, from June 19th to June 25th, 2016. This was the first time that a colloquium of the prestigious and traditional ICGTMP series (which started in 1972 in Marseille, France) took place in South America. (The history of the colloquia can be found at http://icgtmp.blogs.uva.es/)

Author : Andrzej Skowroński,Kunio Yamagata
Publisher : European Mathematical Society
Page : 672 pages
File Size : 51,9 Mb
Release : 2011
Category : Frobenius algebras
ISBN : 3037191023

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Frobenius Algebras by Andrzej Skowroński,Kunio Yamagata Pdf

This is the first of two volumes which will provide a comprehensive introduction to the modern representation theory of Frobenius algebras. The first part of the book serves as a general introduction to basic results and techniques of the modern representation theory of finite dimensional associative algebras over fields, including the Morita theory of equivalences and dualities and the Auslander-Reiten theory of irreducible morphisms and almost split sequences. The second part is devoted to fundamental classical and recent results concerning the Frobenius algebras and their module categories. Moreover, the prominent classes of Frobenius algebras, the Hecke algebras of Coxeter groups, and the finite dimensional Hopf algebras over fields are exhibited. This volume is self contained and the only prerequisite is a basic knowledge of linear algebra. It includes complete proofs of all results presented and provides a rich supply of examples and exercises. The text is primarily addressed to graduate students starting research in the representation theory of algebras as well as mathematicians working in other fields.