Non-commutative differential geometry
type: Item
instance of: scholarly article
title: Non-commutative differential geometry
zbMATH Open document ID: 0592.46056
DOI: 10.1007/BF02698807
published in: Publications Mathématiques
publication date: 1985-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/104008
review text: In this article, Connes lays the groundwork for a theory of noncommutative differential geometry, i.e. differential geometry for noncommutative algebras generalizing the commutative algebra \({\mathcal C}^{\infty}(M)\) of smooth functions on a compact manifold. The idea of doing topology of noncommutative ''topological spaces'', i.e. \(C^*\)- algebras, is not new and has, in fact, found a very satisfactory realization via \(K\)--theory and notably Kasparov's \(KK\)--theory for \(C^*\)- algebras. To do differential geometry, one would like to have an analogue of de Rham (co)homology and characteristic classes. This generalization, the ''cyclic cohomology'' is obtained by Connes in the following way. Let \({\mathfrak A}\) be a complex algebra. The differential envelope \(\Omega\) \({\mathfrak A}\) consists of all linear combinations of abstract (not necessarily antisymmetric) ''differential forms'' \(x_ 0dx_ 1...dx_ n\) over \({\mathfrak A}\) (i.e. \(x_ i\in {\mathfrak A})\) where \(d(xy)=xdy+d(x)y\), x,y\(\in {\mathfrak A}\). In other words \(\Omega\) \({\mathfrak A}\) is the universal algebra containing \({\mathfrak A}\) and admitting a linear map \(d: \Omega{\mathfrak A}\to \Omega{\mathfrak A}\) such that \(d(xy)=xdy+dx y\), \((x,y\in {\mathfrak A})\) and \(d^ 2=0\). Let \(\Omega^ n{\mathfrak A}\) be the space of all such forms of degree \(n\). Every trace \(T: \Omega^ n{\mathfrak A}\to {\mathbb C}\) (\(T\) is a trace if \(T(\omega\omega')= T(\omega'\omega)\) for \(\omega \in \Omega^ k{\mathfrak A}\), \(\omega'\in \Omega^{\ell}{\mathfrak A}\), \(k+\ell =n)\) which is in addition closed, i.e. \(T(d\omega)=0\), \(\forall \omega\), gives rise to an \((n+1)\)-linear functional \(f(x_ 0,...,x_ n)=T(x_ 0dx_ 1...dx_ n)\) which has the following properties (1) \(f(x_ 1,...,x_ n,x_ 0)=(-1)^ nf(x_ 0,...,x_ n)\) (2) \(bf=0\) where \(b\) is the Hochschild boundary operator \(bf(x_ 0,...,x_{n+1})=f(x_ 0x_ 1,...,x_{n+1})-f(x_ 0,x_ 1x_ 2, ...,x_{n+1})+...+(-1)^{n+1}f(x_{n+1}x_ 0,…
Mathematics Subject Classification ID: 46L87
Mathematics Subject Classification ID: 46L89
Mathematics Subject Classification ID: 46L85
Mathematics Subject Classification ID: 58B32
Mathematics Subject Classification ID: 58B34
Mathematics Subject Classification ID: 18G35
Mathematics Subject Classification ID: 18G60
Mathematics Subject Classification ID: 58A12
Mathematics Subject Classification ID: 18F25
Mathematics Subject Classification ID: 19D55
Mathematics Subject Classification ID: 19K35
Mathematics Subject Classification ID: 19K56
Mathematics Subject Classification ID: 58J20
Mathematics Subject Classification ID: 58J22
zbMATH DE Number: 3951325
zbMATH Keywords: noncommutative differential geometry
zbMATH Keywords: characteristic classes
zbMATH Keywords: cyclic cohomology
zbMATH Keywords: differential forms
zbMATH Keywords: trace
zbMATH Keywords: Hochschild boundary operator
zbMATH Keywords: subcomplex
zbMATH Keywords: Hochschild complex
zbMATH Keywords: Ext-theory
zbMATH Keywords: Chern-character
zbMATH Keywords: \(K\)--homology
zbMATH Keywords: Kasparov's KK-theory
zbMATH Keywords: Schatten class
zbMATH Keywords: de Rham homology group
zbMATH Keywords: long exact sequence
zbMATH Keywords: irrational rotation algebra
Wikidata QID: Q56226487
author: Alain Connes
reviewed by: Joachim Cuntz
MaRDI profile type: Publication
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