An Introduction to Noncommutative Spaces and Their Geometries: 51 (Lecture Notes in Physics) - Hardcover

Landi, Giovanni

Hardcover

ISBN 10: 3540635092 ISBN 13: 9783540635093

Publisher: Springer-Verlag Berlin and Heidelberg GmbH & Co. K, 1997

Synopsis

These lecture notes are an introduction to several ideas and applications of noncommutative geometry. It starts with a not necessarily commutative but associative algebra which is thought of as the algebra of functions on some 'virtual noncommutative space'. Attention is switched from spaces, which in general do not even exist, to algebras of functions. In these notes, particular emphasis is put on seeing noncommutative spaces as concrete spaces, namely as a collection of points with a topology. The necessary mathematical tools are presented in a systematic and accessible way and include among other things, C'*-algebras, module theory and K-theory, spectral calculus, forms and connection theory. Application to Yang--Mills, fermionic, and gravity models are described. Also the spectral action and the related invariance under automorphism of the algebra is illustrated. Some recent work on noncommutative lattices is presented. These lattices arose as topologically nontrivial approximations to 'contuinuum' topological spaces. They have been used to construct quantum-mechanical and field-theory models, alternative models to lattice gauge theory, with nontrivial topological content. This book will be essential to physicists and mathematicians with an interest in noncommutative geometry and its uses in physics.

About this title

Synopsis

These notes arose from a series of introductory seminars on noncommutative geometry the author gave at the University of Trieste. They are mainly an introduction to Connes' noncommutative geometry and could serve as a "first aid kit" before one ventures into the landscape of Connes' theory. Emphasis is placed on noncommutative spaces seen as concrete spaces. Examples of noncommutative spaces are provided by noncommutative lattices. The latter are the subject of work the author is doing in collaboration with others. These notes are also meant to be an introduction to these researches.

Edition details

• Publisher: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
• Publication date: 1997
• ISBN 10: 3540635092
• ISBN 13: 9783540635093
• Binding: Hardcover
• Language: English
• Number of pages: 222