Groupoids in Analysis, Geometry, and Physics
Title | Groupoids in Analysis, Geometry, and Physics PDF eBook |
Author | Arlan Ramsay |
Publisher | American Mathematical Soc. |
Pages | 208 |
Release | 2001 |
Genre | Mathematics |
ISBN | 0821820427 |
Groupoids often occur when there is symmetry of a nature not expressible in terms of groups. Other uses of groupoids can involve something of a dynamical nature. Indeed, some of the main examples come from group actions. It should also be noted that in many situations where groupoids have been used, the main emphasis has not been on symmetry or dynamics issues. While the implicit symmetry and dynamics are relevant, the groupoid records mostly the structure of the space of leaves and the holonomy. More generally, the use of groupoids is very much related to various notions of orbit equivalance. This book presents the proceedings from the Joint Summer Research Conference on ``Groupoids in Analysis, Geometry, and Physics'' held in Boulder, CO. The book begins with an introduction to ways in which groupoids allow a more comprehensive view of symmetry than is seen via groups. Topics range from foliations, pseudo-differential operators, $KK$-theory, amenability, Fell bundles, and index theory to quantization of Poisson manifolds. Readers will find examples of important tools for working with groupoids. This book is geared to students and researchers. It is intended to improve their understanding of groupoids and to encourage them to look further while learning about the tools used.
Groupoids, Inverse Semigroups, and their Operator Algebras
Title | Groupoids, Inverse Semigroups, and their Operator Algebras PDF eBook |
Author | Alan Paterson |
Publisher | Springer Science & Business Media |
Pages | 286 |
Release | 2012-12-06 |
Genre | Mathematics |
ISBN | 1461217741 |
In recent years, it has become increasingly clear that there are important connections relating three concepts -- groupoids, inverse semigroups, and operator algebras. There has been a great deal of progress in this area over the last two decades, and this book gives a careful, up-to-date and reasonably extensive account of the subject matter. After an introductory first chapter, the second chapter presents a self-contained account of inverse semigroups, locally compact and r-discrete groupoids, and Lie groupoids. The section on Lie groupoids in chapter 2 contains a detailed discussion of groupoids particularly important in noncommutative geometry, including the holonomy groupoids of a foliated manifold and the tangent groupoid of a manifold. The representation theories of locally compact and r-discrete groupoids are developed in the third chapter, and it is shown that the C*-algebras of r-discrete groupoids are the covariance C*-algebras for inverse semigroup actions on locally compact Hausdorff spaces. A final chapter associates a universal r-discrete groupoid with any inverse semigroup. Six subsequent appendices treat topics related to those covered in the text. The book should appeal to a wide variety of professional mathematicians and graduate students in fields such as operator algebras, analysis on groupoids, semigroup theory, and noncommutative geometry. It will also be of interest to mathematicians interested in tilings and theoretical physicists whose focus is modeling quasicrystals with tilings. An effort has been made to make the book lucid and 'user friendly"; thus it should be accessible to any reader with a basic background in measure theory and functional analysis.
Poisson Geometry, Deformation Quantisation and Group Representations
Title | Poisson Geometry, Deformation Quantisation and Group Representations PDF eBook |
Author | Simone Gutt |
Publisher | Cambridge University Press |
Pages | 380 |
Release | 2005-06-21 |
Genre | Mathematics |
ISBN | 9780521615051 |
An accessible introduction to Poisson geometry suitable for graduate students.
Group Representations, Ergodic Theory, and Mathematical Physics
Title | Group Representations, Ergodic Theory, and Mathematical Physics PDF eBook |
Author | Robert S. Doran |
Publisher | American Mathematical Soc. |
Pages | 458 |
Release | 2008 |
Genre | Mathematics |
ISBN | 0821842250 |
George Mackey was an extraordinary mathematician of great power and vision. His profound contributions to representation theory, harmonic analysis, ergodic theory, and mathematical physics left a rich legacy for researchers that continues today. This book is based on lectures presented at an AMS special session held in January 2007 in New Orleans dedicated to his memory. The papers, written especially for this volume by internationally-known mathematicians and mathematical physicists, range from expository and historical surveys to original high-level research articles. The influence of Mackey's fundamental ideas is apparent throughout. The introductory article contains recollections from former students, friends, colleagues, and family as well as a biography describing his distinguished career as a mathematician at Harvard, where he held the Landon D. Clay Professorship of Mathematics.
An Introduction to Groups, Groupoids and Their Representations
Title | An Introduction to Groups, Groupoids and Their Representations PDF eBook |
Author | Alberto Ibort |
Publisher | CRC Press |
Pages | 242 |
Release | 2019-10-28 |
Genre | Mathematics |
ISBN | 1351869566 |
This book offers an introduction to the theory of groupoids and their representations encompassing the standard theory of groups. Using a categorical language, developed from simple examples, the theory of finite groupoids is shown to knit neatly with that of groups and their structure as well as that of their representations is described. The book comprises numerous examples and applications, including well-known games and puzzles, databases and physics applications. Key concepts have been presented using only basic notions so that it can be used both by students and researchers interested in the subject. Category theory is the natural language that is being used to develop the theory of groupoids. However, categorical presentations of mathematical subjects tend to become highly abstract very fast and out of reach of many potential users. To avoid this, foundations of the theory, starting with simple examples, have been developed and used to study the structure of finite groups and groupoids. The appropriate language and notions from category theory have been developed for students of mathematics and theoretical physics. The book presents the theory on the same level as the ordinary and elementary theories of finite groups and their representations, and provides a unified picture of the same. The structure of the algebra of finite groupoids is analysed, along with the classical theory of characters of their representations. Unnecessary complications in the formal presentation of the subject are avoided. The book offers an introduction to the language of category theory in the concrete setting of finite sets. It also shows how this perspective provides a common ground for various problems and applications, ranging from combinatorics, the topology of graphs, structure of databases and quantum physics.
Lie Groupoids and Lie Algebroids in Differential Geometry
Title | Lie Groupoids and Lie Algebroids in Differential Geometry PDF eBook |
Author | K. Mackenzie |
Publisher | Cambridge University Press |
Pages | 345 |
Release | 1987-06-25 |
Genre | Mathematics |
ISBN | 052134882X |
This book provides a striking synthesis of the standard theory of connections in principal bundles and the Lie theory of Lie groupoids. The concept of Lie groupoid is a little-known formulation of the concept of principal bundle and corresponding to the Lie algebra of a Lie group is the concept of Lie algebroid: in principal bundle terms this is the Atiyah sequence. The author's viewpoint is that certain deep problems in connection theory are best addressed by groupoid and Lie algebroid methods. After preliminary chapters on topological groupoids, the author gives the first unified and detailed account of the theory of Lie groupoids and Lie algebroids. He then applies this theory to the cohomology of Lie algebroids, re-interpreting connection theory in cohomological terms, and giving criteria for the existence of (not necessarily Riemannian) connections with prescribed curvature form. This material, presented in the last two chapters, is work of the author published here for the first time. This book will be of interest to differential geometers working in general connection theory and to researchers in theoretical physics and other fields who make use of connection theory.
An Introduction to C*-Algebras and Noncommutative Geometry
Title | An Introduction to C*-Algebras and Noncommutative Geometry PDF eBook |
Author | Heath Emerson |
Publisher | Springer Nature |
Pages | 548 |
Release | |
Genre | |
ISBN | 3031598504 |