Abelian, in mathematics, is used in many different definitions, named after Norwegian mathematician Niels Henrik Abel: In group theory:

Abelian group, a group in which the binary operation is commutative

Abelian function, a meromorphic function on an abelian variety

Abelian variety, a complex torus that can be embedded into projective space

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

Abelianisation summary and related information.

Abelian category, a preabelian category in which morphisms and objects can be added.

Preabelian category is an additive category that has all kernels and cokernels

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

Category of abelian groups Ab has abelian groups as objects and group homomorphisms as morphisms

Abelian integral, a function related to the indefinite integral of a differential of the first kind

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

Metabelian group is a group where the commutator subgroup is abelian