Noncommutative geometry

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In Mathematics, Noncommutative geometry, or NCG, is concerned with the possible spatial interpretations of algebraic structures for which the commutative law fails; that is, for which xy does not always equal yx. For example; 3 steps of 4 units and 4 steps of 3 units length might be different in noncommutative spaces. Although one could technically construct geometries by simply removing this condition (commutativity), the results are typically trivial or uninteresting. The most common usage of the term, therefore, refers to what is properly called Differential Noncommutative Geometry, a subject which was developed extensively by French mathematician Alain Connes. The challenge of NCG theory is to get around the lack of commutative multiplication, which is a requirement of previous geometric theories of algebraic structures.

1 Motivation
2 Non-commutative C*-algebras, von Neumann algebras
3 Non-commutative differentiable manifolds
4 Non-commutative affine schemes
5 Examples of non-commutative spaces
6 History
7 See also
8 External links

In mathematics, there is a close relationship between spaces, which are geometric in nature, and the numerical functions on them. In general, such functions will form a commutative ring. For instance, one may take the ring C(X) of continuous complex-valued functions on a topological space X. In many important cases (e.g., if X is a compact Hausdorff space), we can recover X from C(X), and therefore it makes some sense to say that X has commutative geometry.

For other cases and applications, including mathematical physics¹ and functional analysis, non-commutative rings arise as the natural candidates for a ring of functions on some non-commutative "space". "Non-commutative spaces", however defined, cannot be too similar to ordinary topological spaces, as these are known to correspond to commutative rings in many important cases. For this reason, the field is also called non-commutative topology — some of the motivating questions of the theory are concerned with extending known topological invariants to these new spaces. That is, the "space" itself is used as some sort of middle term.

Non-commutative C*-algebras are often now called non-commutative spaces. This is by analogy with the Gelfand representation, which shows that commutative C*-algebras are dual to locally compact Hausdorff spaces. In general, one can associate to any C*-algebra S a topological space Ŝ; see spectrum of a C*-algebra.

For the duality between σ-finite measure spaces and commutative von Neumann algebras, noncommutative von Neumann algebras are called non-commutative measure spaces.

Another area of study is that of non-commutative differentiable manifolds. An ordinary differentiable manifold can be characterized by the commutative algebra of smooth functions defined on the manifold, and the space of smooth sections of its tangent bundle, cotangent bundle and other fiber bundles. All these spaces are modules over the commutative algebra of smooth functions. The concepts of exterior derivative, Lie derivative and covariant derivative are also important elements in understanding derivations over this algebra. In the non-commutative case, the algebras in question are non-commutative. To handle differential forms, one must work with the graded exterior algebra bundle of all p-forms under the wedge product and look at its algebra of smooth sections. A "differential" is taken to be an antiderivation (or something more general) on this algebra which increases the grading by 1 and is quadratically nilpotent.

In analogy to the duality between affine schemes and commutative rings, we can also have noncommutative affine schemes. For example, there exist an analog of the celebrated Serre duality for noncommutative projective schemes. [1]

In Weyl quantization, the symplectic phase space of classical mechanics is deformed into a non-commutative phase space generated by the position and momentum operators.

The standard model of particle physics is another example of a noncommutative geometry, cf noncommutative standard model.

Some of the theory developed by Alain Connes to handle noncommutative geometry at a technical level has roots in older attempts, in particular in ergodic theory. The proposal of George Mackey to create a virtual subgroup theory, with respect to which ergodic group actions would become homogeneous spaces of an extended kind, has by now been subsumed.