Abelian

Categories: Disambiguation

In mathematics, the term abelian is used in many different definitions:

Contents 1 In group theory 2 In Galois theory 3 In real analysis 4 In functional analysis 5 In topology and number theory 6 In category theory

In group theory

• An abelian group is a group in which the binary operation is commutative.
• The category of abelian groups Ab has abelian groups as objects and group homomorphisms as morphisms.
• A metabelian group is a group where the commutator subgroup is contained in the center.
• Any group is "made abelian" by its abelianisation.

In Galois theory

• An abelian extension is a field extension for which the associated Galois group is abelian.

In real analysis

• Abelian theorems are used in the summation of divergent series.

In functional analysis

• An abelian von Neumann algebra is a von Neumann algebra of operators on a Hilbert space in which all elements commute.

In topology and number theory

• An abelian variety is a complex torus that can be embedded into projective space. This includes the elliptic curves and the abelian varieties of CM-type.
• An abelian function is a meromorphic function on an abelian variety.
• An abelian integral is a function related to the indefinite integral of a differential of the first kind. This includes the elliptic integrals.

In category theory

• A preabelian category is an additive category that has all kernels and cokernels.
• An abelian category is a preabelian category in which morphisms and objects can be added.