An Introduction to Noncommutative Spaces and Their Geometries. 1st ed. 1997

種類:

電子ブック

責任表示:

by Giovanni Landi

出版情報:

Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 1997

著者名:

• Landi, Giovanni.
• SpringerLink (Online service)

シリーズ名:

Lecture Notes in Physics Monographs ; 51

ISBN:

9783540149491 [354014949X]    

注記:

Noncommutative Spaces and Algebras of Functions -- Projective Systems of Noncommutative Lattices -- Modules as Bundles -- A Few Elements of K-Theory -- The Spectral Calculus -- Noncommutative Differential Forms -- Connections on Modules -- Field Theories on Modules -- Gravity Models -- Quantum Mechanical Models on Noncommutative Lattices.
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' topolo

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