Noncommutative geometry

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In mathematics, non-commutative geometry 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. The challenge of the theory is to get around the lack of commutative multiplication, which is a requirement of previous geometric theories of such structures.

Motivation

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, von Neumann algebras

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 locally compact measure spaces and commutative von Neumann algebras, noncommutative von Neumann algebras are called non-commutative measure spaces.

Non-commutative differentiable manifolds

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.

Non-commutative affine schemes

In analogy to the duality between affine schemes and commutative rings, we can also have noncommutative affine schemes.

Examples of non-commutative spaces

History

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.

See also


[1] The applications in particle physics are described on the entry for Noncommutative quantum field theory

es:Geometría no conmutativa fr:Géométrie non commutative