Noncommutative geometry

Noncommutative geometry (NCG) is a branch of mathematics that studies geometric ideas through noncommutative algebras. In ordinary geometry, a space can often be studied by means of a commutative algebra of functions on it; noncommutative geometry extends this viewpoint to algebras in which the product of two elements need not commute. Such algebras are treated as analogues of algebras of functions on generalized, or "noncommutative", spaces.

The subject is not a single formalism. It includes operator-algebraic methods based on C*-algebras, von Neumann algebras, and spectral triples; algebraic approaches to noncommutative rings and graded algebras; and constructions related to deformation quantization, groupoid C*-algebras, cyclic homology, and K-theory. A standard example is the noncommutative torus, whose algebra is generated by two unitary elements satisfying a twisted commutation relation and which has served as a test case for noncommutative versions of vector bundles, connections, curvature and index theory.

This article is issued from Wikipedia. The text is available under Creative Commons Attribution-Share Alike 4.0 unless otherwise noted. Additional terms may apply for the media files.