Ickeabeliansk

Ickeabeliansk is a Swedish term meaning "non-abelian." In mathematics and physics it is used to describe objects for which the operation is not commutative; that is, there exist elements a and b such that ab ≠ ba.

In abstract algebra, the primary focus is on non-abelian groups. A group is non-abelian when the binary operation is not commutative. Classic examples include the symmetric group S3, the dihedral group D4, and the general linear group GL(n, F) for n ≥ 2. Non-abelian structures recur in representation theory, topology and geometry, where their rich symmetries lead to nontrivial representations and actions. By contrast, abelian structures have the property that all elements commute, such as the integers under addition.

In physics, non-abelian concepts appear in gauge theories, where the gauge group is non-commutative. The Standard Model uses non-abelian groups such as SU(3) for color and SU(2) for weak isospin. In two-dimensional systems, non-abelian anyons exhibit non-commuting braiding operations, a feature explored in proposals for topological quantum computation.

Origin of the term: "icke" means "not" and "abeliansk" derives from Abelian, named after Niels Henrik Abel. The term signals the non-commutative character that distinguishes these structures from abelian ones. The concept is central across mathematics and physics, highlighting the role of non-commutativity in symmetry and interaction.