Abelian

For the north-African Christian sect, see Abelians.

In mathematics, Abelian refers to any of number of different mathematical concepts named after Niels Henrik Abel:

Group theory

• Abelian group, a group in which the binary operation is commutative
  • Category of abelian groups Ab has abelian groups as objects and group homomorphisms as morphisms
• Metabelian group is a group where the commutator subgroup is abelian
• Abelianisation

Galois theory

• Abelian extension, a field extension for which the associated Galois group is abelian

Real analysis

• Abelian and tauberian theorems, used in the summation of divergent series

Functional analysis

• Abelian von Neumann algebra, a von Neumann algebra of operators on a Hilbert space in which all elements commute

Topology and number theory

• Abelian variety, a complex torus that can be embedded into projective space
• Abelian surface, a two-dimensional abelian variety
• Abelian function, a meromorphic function on an abelian variety
• Abelian integral, a function related to the indefinite integral of a differential of the first kind

In category theory

• Pre-abelian category, an additive category that has all kernels and cokernels
• Abelian category, a preabelian category in which every monomorphism is a kernel and every epimorphism is a cokernel

In physics

• A gauge theory is abelian or non-abelian depending on whether its symmetry group is commutative or non-commutative, respectively

Other uses

• Abelians, 4th-century Christian sect
• Hovhannes Abelian (1865–1936), Armenian actor