Algebraic Foundations Of Non Commutative Differential Geometry And Quantum Groups

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Algebraic Foundations of Non-Commutative Differential Geometry and Quantum Groups

Ludwig Pittner
Algebraic Foundations of Non Commutative Differential Geometry and Quantum Groups Book PDF

Author : Ludwig Pittner
Publisher : Springer Science & Business Media
Page : 476 pages
File Size : 54,8 Mb
Release : 2009-01-29
Category : Science
ISBN : 9783540478010

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Algebraic Foundations of Non-Commutative Differential Geometry and Quantum Groups by Ludwig Pittner Pdf

Quantum groups and quantum algebras as well as non-commutative differential geometry are important in mathematics and considered to be useful tools for model building in statistical and quantum physics. This book, addressing scientists and postgraduates, contains a detailed and rather complete presentation of the algebraic framework. Introductory chapters deal with background material such as Lie and Hopf superalgebras, Lie super-bialgebras, or formal power series. Great care was taken to present a reliable collection of formulae and to unify the notation, making this volume a useful work of reference for mathematicians and mathematical physicists.

Quantum Groups and Noncommutative Spaces

Matilde Marcolli,Deepak Parashar
Quantum Groups and Noncommutative Spaces Book PDF

Author : Matilde Marcolli,Deepak Parashar
Publisher : Springer Science & Business Media
Page : 247 pages
File Size : 42,6 Mb
Release : 2010-11-02
Category : Mathematics
ISBN : 9783834898319

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Quantum Groups and Noncommutative Spaces by Matilde Marcolli,Deepak Parashar Pdf

This book is aimed at presenting different methods and perspectives in the theory of Quantum Groups, bridging between the algebraic, representation theoretic, analytic, and differential-geometric approaches. It also covers recent developments in Noncommutative Geometry, which have close relations to quantization and quantum group symmetries. The volume collects surveys by experts which originate from an acitvity at the Max-Planck-Institute for Mathematics in Bonn.

Quantum Groups and Noncommutative Geometry

Yuri I. Manin
Quantum Groups and Noncommutative Geometry Book PDF

Author : Yuri I. Manin
Publisher : Springer
Page : 125 pages
File Size : 46,9 Mb
Release : 2018-10-11
Category : Mathematics
ISBN : 9783319979878

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Quantum Groups and Noncommutative Geometry by Yuri I. Manin Pdf

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.

Quantum Groups, Noncommutative Geometry and Fundamental Physical Interactions

Daniel Kastler
Quantum Groups Noncommutative Geometry and Fundamental Physical Interactions Book PDF

Author : Daniel Kastler
Publisher : Unknown
Page : 490 pages
File Size : 54,8 Mb
Release : 1999
Category : Mathematics
ISBN : STANFORD:36105023609014

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Quantum Groups, Noncommutative Geometry and Fundamental Physical Interactions by Daniel Kastler Pdf

Contents include: Hochschild Homology of Function Algebras Associated with Singularities; On the KK-Theory of Stable Projective Limits; Noncommutative Integrability; Gauge Invariance of the Chern-Simons Action in Noncommutative Geometry; The Analysis of the Hochshild Homology; Coproducts and Operations on Cyclic Cohomology; Powers of Quantum Matrices and Relations Between Them; Introductory Notes on Extensions of Hopf Algebras; Hopf Algebras from the Quantum Geometry Point of View; Equation Pentagonale, Bige bres et Espaces de Modules; Chiral Anomalies in the Spectral Action; Standard Model and Unimodularity Condition; On Feynman Graphs as Elements of a Hopf Algebra.

Quantum Riemannian Geometry

Edwin J. Beggs,Shahn Majid
Quantum Riemannian Geometry Book PDF

Author : Edwin J. Beggs,Shahn Majid
Publisher : Springer Nature
Page : 809 pages
File Size : 43,7 Mb
Release : 2020-01-31
Category : Science
ISBN : 9783030302948

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Quantum Riemannian Geometry by Edwin J. Beggs,Shahn Majid Pdf

This book provides a comprehensive account of a modern generalisation of differential geometry in which coordinates need not commute. This requires a reinvention of differential geometry that refers only to the coordinate algebra, now possibly noncommutative, rather than to actual points. Such a theory is needed for the geometry of Hopf algebras or quantum groups, which provide key examples, as well as in physics to model quantum gravity effects in the form of quantum spacetime. The mathematical formalism can be applied to any algebra and includes graph geometry and a Lie theory of finite groups. Even the algebra of 2 x 2 matrices turns out to admit a rich moduli of quantum Riemannian geometries. The approach taken is a `bottom up’ one in which the different layers of geometry are built up in succession, starting from differential forms and proceeding up to the notion of a quantum `Levi-Civita’ bimodule connection, geometric Laplacians and, in some cases, Dirac operators. The book also covers elements of Connes’ approach to the subject coming from cyclic cohomology and spectral triples. Other topics include various other cohomology theories, holomorphic structures and noncommutative D-modules. A unique feature of the book is its constructive approach and its wealth of examples drawn from a large body of literature in mathematical physics, now put on a firm algebraic footing. Including exercises with solutions, it can be used as a textbook for advanced courses as well as a reference for researchers.

Quantum Isometry Groups

Debashish Goswami,Jyotishman Bhowmick
Quantum Isometry Groups Book PDF

Author : Debashish Goswami,Jyotishman Bhowmick
Publisher : Springer
Page : 235 pages
File Size : 54,6 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.

Noncommutative Geometry

Igor V. Nikolaev
Noncommutative Geometry Book PDF

Author : Igor V. Nikolaev
Publisher : Walter de Gruyter GmbH & Co KG
Page : 403 pages
File Size : 40,9 Mb
Release : 2017-11-07
Category : Mathematics
ISBN : 9783110543483

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Noncommutative Geometry by Igor V. Nikolaev Pdf

This book covers the basics of noncommutative geometry (NCG) and its applications in topology, algebraic geometry, and number theory. The author takes up the practical side of NCG and its value for other areas of mathematics. A brief survey of the main parts of NCG with historical remarks, bibliography, and a list of exercises is included. The presentation is intended for graduate students and researchers with interests in NCG, but will also serve nonexperts in the field. Contents Part I: Basics Model examples Categories and functors C∗-algebras Part II: Noncommutative invariants Topology Algebraic geometry Number theory Part III: Brief survey of NCG Finite geometries Continuous geometries Connes geometries Index theory Jones polynomials Quantum groups Noncommutative algebraic geometry Trends in noncommutative geometry

Introduction to Quantum Groups

Masud Chaichian,Andrei Pavlovich Demichev
Introduction to Quantum Groups Book PDF

Author : Masud Chaichian,Andrei Pavlovich Demichev
Publisher : World Scientific
Page : 362 pages
File Size : 51,8 Mb
Release : 1996
Category : Science
ISBN : 9810226233

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Introduction to Quantum Groups by Masud Chaichian,Andrei Pavlovich Demichev Pdf

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.

Perspectives on Noncommutative Geometry

Masoud Khalkhali
Perspectives on Noncommutative Geometry Book PDF

Author : Masoud Khalkhali
Publisher : American Mathematical Soc.
Page : 176 pages
File Size : 41,7 Mb
Release : 2011
Category : Algebra, Homological
ISBN : 9780821848494

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Perspectives on Noncommutative Geometry by Masoud Khalkhali Pdf

This volume represents the proceedings of the Noncommutative Geometry Workshop that was held as part of the thematic program on operator algebras at the Fields Institute in May 2008. Pioneered by Alain Connes starting in the late 1970s, noncommutative geometry was originally inspired by global analysis, topology, operator algebras, and quantum physics. Its main applications were to settle some long-standing conjectures, such as the Novikov conjecture and the Baum-Connes conjecture. Next came the impact of spectral geometry and the way the spectrum of a geometric operator, like the Laplacian, holds information about the geometry and topology of a manifold, as in the celebrated Weyl law. This has now been vastly generalized through Connes' notion of spectral triples. Finally, recent years have witnessed the impact of number theory, algebraic geometry and the theory of motives, and quantum field theory on noncommutative geometry. Almost all of these aspects are touched upon with new results in the papers of this volume. This book is intended for graduate students and researchers in both mathematics and theoretical physics who are interested in noncommutative geometry and its applications.

Quantum Field Theory and Noncommutative Geometry

Ursula Carow-Watamura,Yoshiaki Maeda
Quantum Field Theory and Noncommutative Geometry Book PDF

Author : Ursula Carow-Watamura,Yoshiaki Maeda
Publisher : Springer Science & Business Media
Page : 316 pages
File Size : 46,7 Mb
Release : 2005-02-21
Category : Mathematics
ISBN : 3540239006

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Quantum Field Theory and Noncommutative Geometry by Ursula Carow-Watamura,Yoshiaki Maeda Pdf

This volume reflects the growing collaboration between mathematicians and theoretical physicists to treat the foundations of quantum field theory using the mathematical tools of q-deformed algebras and noncommutative differential geometry. A particular challenge is posed by gravity, which probably necessitates extension of these methods to geometries with minimum length and therefore quantization of space. This volume builds on the lectures and talks that have been given at a recent meeting on "Quantum Field Theory and Noncommutative Geometry." A considerable effort has been invested in making the contributions accessible to a wider community of readers - so this volume will not only benefit researchers in the field but also postgraduate students and scientists from related areas wishing to become better acquainted with this field.

Hopf Algebras in Noncommutative Geometry and Physics

Stefaan Caenepeel,Fred Van Oystaeyen
Hopf Algebras in Noncommutative Geometry and Physics Book PDF

Author : Stefaan Caenepeel,Fred Van Oystaeyen
Publisher : CRC Press
Page : 344 pages
File Size : 51,6 Mb
Release : 2019-05-07
Category : Mathematics
ISBN : 9781482276718

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Hopf Algebras in Noncommutative Geometry and Physics by Stefaan Caenepeel,Fred Van Oystaeyen Pdf

This comprehensive reference summarizes the proceedings and keynote presentations from a recent conference held in Brussels, Belgium. Offering 1155 display equations, this volume contains original research and survey papers as well as contributions from world-renowned algebraists. It focuses on new results in classical Hopf algebras as well as the

Noncommutative Algebraic Geometry and Representations of Quantized Algebras

A. Rosenberg
Noncommutative Algebraic Geometry and Representations of Quantized Algebras Book PDF

Author : A. Rosenberg
Publisher : Springer Science & Business Media
Page : 333 pages
File Size : 45,6 Mb
Release : 2013-03-09
Category : Mathematics
ISBN : 9789401584302

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Noncommutative Algebraic Geometry and Representations of Quantized Algebras by A. Rosenberg Pdf

This book is based on lectures delivered at Harvard in the Spring of 1991 and at the University of Utah during the academic year 1992-93. Formally, the book assumes only general algebraic knowledge (rings, modules, groups, Lie algebras, functors etc.). It is helpful, however, to know some basics of algebraic geometry and representation theory. Each chapter begins with its own introduction, and most sections even have a short overview. The purpose of what follows is to explain the spirit of the book and how different parts are linked together without entering into details. The point of departure is the notion of the left spectrum of an associative ring, and the first natural steps of general theory of noncommutative affine, quasi-affine, and projective schemes. This material is presented in Chapter I. Further developments originated from the requirements of several important examples I tried to understand, to begin with the first Weyl algebra and the quantum plane. The book reflects these developments as I worked them out in reallife and in my lectures. In Chapter 11, we study the left spectrum and irreducible representations of a whole lot of rings which are of interest for modern mathematical physics. The dasses of rings we consider indude as special cases: quantum plane, algebra of q-differential operators, (quantum) Heisenberg and Weyl algebras, (quantum) enveloping algebra ofthe Lie algebra sl(2) , coordinate algebra of the quantum group SL(2), the twisted SL(2) of Woronowicz, so called dispin algebra and many others.

Quantum Groups and Their Representations

Anatoli Klimyk,Konrad Schmüdgen
Quantum Groups and Their Representations Book PDF

Author : Anatoli Klimyk,Konrad Schmüdgen
Publisher : Springer Science & Business Media
Page : 568 pages
File Size : 45,9 Mb
Release : 2012-12-06
Category : Science
ISBN : 9783642608964

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Quantum Groups and Their Representations by Anatoli Klimyk,Konrad Schmüdgen Pdf

This book start with an introduction to quantum groups for the beginner and continues as a textbook for graduate students in physics and in mathematics. It can also be used as a reference by more advanced readers. The authors cover a large but well-chosen variety of subjects from the theory of quantum groups (quantized universal enveloping algebras, quantized algebras of functions) and q-deformed algebras (q-oscillator algebras), their representations and corepresentations, and noncommutative differential calculus. The book is written with potential applications in physics and mathematics in mind. The basic quantum groups and quantum algebras and their representations are given in detail and accompanied by explicit formulas. A number of topics and results from the more advanced general theory are developed and discussed.

Commutative Algebra and Noncommutative Algebraic Geometry

David Eisenbud,Srikanth B. Iyengar,Anurag K. Singh,J. Toby Stafford,Michel Van den Bergh
Commutative Algebra and Noncommutative Algebraic Geometry Book PDF

Author : David Eisenbud,Srikanth B. Iyengar,Anurag K. Singh,J. Toby Stafford,Michel Van den Bergh
Publisher : Cambridge University Press
Page : 463 pages
File Size : 45,6 Mb
Release : 2015-11-19
Category : Mathematics
ISBN : 9781107065628

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Commutative Algebra and Noncommutative Algebraic Geometry by David Eisenbud,Srikanth B. Iyengar,Anurag K. Singh,J. Toby Stafford,Michel Van den Bergh Pdf

This book surveys fundamental current topics in these two areas of research, emphasising the lively interaction between them. Volume 1 contains expository papers ideal for those entering the field.

Noncommutative Geometry

Alain Connes,Joachim Cuntz,Erik G. Guentner,Nigel Higson,Jerome Kaminker,John E. Roberts
Noncommutative Geometry Book PDF

Author : Alain Connes,Joachim Cuntz,Erik G. Guentner,Nigel Higson,Jerome Kaminker,John E. Roberts
Publisher : Springer Science & Business Media
Page : 372 pages
File Size : 43,8 Mb
Release : 2003-12-08
Category : Mathematics
ISBN : 3540203575

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Noncommutative Geometry by Alain Connes,Joachim Cuntz,Erik G. Guentner,Nigel Higson,Jerome Kaminker,John E. Roberts Pdf

Noncommutative Geometry is one of the most deep and vital research subjects of present-day Mathematics. Its development, mainly due to Alain Connes, is providing an increasing number of applications and deeper insights for instance in Foliations, K-Theory, Index Theory, Number Theory but also in Quantum Physics of elementary particles. The purpose of the Summer School in Martina Franca was to offer a fresh invitation to the subject and closely related topics; the contributions in this volume include the four main lectures, cover advanced developments and are delivered by prominent specialists.