Overview

Non-commutative geometry is developed in main as a generalization of the calculus in commutative rings of smooth functions [107; 194; 267; 290]. Accordingly, a non-commutative generalization of differential geometry is phrased in terms of the differential calculus over a non-commutative ring which replaces the exterior algebra of differential forms. The Chevalley-Eilenberg differential calculus over a commutative ring in Section 1.6 is straightforwardly generalized to a non-commutative -ring . However, the extension of the notion of a differential operator in Section 1.2 to modules over a non-commutative ring meets difficulties (see Section 8.3). In a general setting, any non-commutative ring can be called into play, but one often follows the more deep analogy to the case of commutative smooth function rings. In Connes' commutative geometry, is the algebra ? ^?( X) of smooth complex functions on a compact manifold X. It is a dense subalgebra of the C*-algebra of continuous complex functions on X. Generalizing this case, Connes' non-commutative geometry [107; 109] addresses the differential calculus over an involutive algebra of bounded operators in a Hilbert space E and, furthermore, studies a representation of this differential calculus by operators in E (see Section 8.5). Section 8.6 is devoted to another variant of non-commutative geometry, where one assigns to a Poisson manifold coming from a Lie groupoid the C*-algebra of (see Section 3.4). The key point is that these assignments are functorial if one considers certain categories of...