An introduction to noncommutative spaces and their geometries / Giovanni Landi.

Format:

Book

Author/Creator:

Landi, Giovanni, 1959- author.

Series:

Lecture Notes in Physics Monographs, 0940-7677 ; 51

Language:

English

Subjects (All):

Noncommutative differential geometry.

Physical Description:

1 online resource (XV, 207 p.)

Edition:

1st ed. 1997.

Place of Publication:

Berlin ; Heidelberg : Springer, [1997]

Language Note:

English

Summary:

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.

Contents:

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.

Notes:

Bibliographic Level Mode of Issuance: Monograph

Includes bibliographical references and index.

Description based on print version record.

ISBN:

3-540-14949-X