noncommutative

Noncommutative refers to mathematical structures in which the order of a binary operation matters. In particular, a noncommutative algebra contains elements a and b with ab ≠ ba. By contrast, commutative algebras satisfy ab = ba for all elements. Common examples include the algebra of n-by-n matrices over a field, where matrix multiplication is generally noncommutative, and the quaternions, which satisfy ij = k while ji = -k.

Noncommutative algebra studies such structures—noncommutative rings, algebras, and their representations. A broader theme is to generalize geometric and analytic notions to noncommutative settings. Noncommutative geometry, pioneered by Alain Connes, uses operator algebras as noncommuting analogues of function algebras on spaces. C*-algebras and von Neumann algebras provide analytic frameworks for noncommutative phenomena.

Noncommutative probability extends classical probability to noncommuting random variables represented by operators, with expectations given by traces or states on algebras. In physics, noncommutativity of observables is fundamental: positions and momenta satisfy the Heisenberg commutation relation, leading to the uncertainty principle and quantum mechanics.

Applications and related areas include representation theory, deformation theory, quantum groups, and topological invariants arising from quantum methods. The term noncommutative thus signals a departure from commutative intuition and invites algebraic, analytic, and geometric techniques to study systems where order matters.