Abelian

From Wikipedia, the free encyclopedia

Abelian, in mathematics, is used in many different definitions:

In 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

In Galois theory:

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

In real analysis:

• Abelian theorems, used in the summation of divergent series

In functional analysis

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

In topology and number theory

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

• Preabelian category is an additive category that has all kernels and cokernels
• Abelian category, a preabelian category in which morphisms and objects can be added.