An Introduction to Noncommutative Spaces and Their Geometries

Giovanni Landi

ISBN 10: 3662141094 / ISBN 13: 9783662141090

Editore: Springer Berlin Heidelberg Aug 2014, 2014

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This item is printed on demand - it takes 3-4 days longer - Neuware -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. 224 pp. Englisch.

Codice articolo 9783662141090

Dati bibliografici

• Titolo: An Introduction to Noncommutative Spaces and Their Geometries
• Editore: Springer Berlin Heidelberg Aug 2014
• Data di pubblicazione: 2014
• Lingua: Inglese
• ISBN 10: 3662141094
• ISBN 13: 9783662141090
• Rilegatura: Taschenbuch
• Condizione: Neu

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Riassunto

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.