Abelian
Encyclopedia
In mathematics, Abelian refers to any of number of different mathematical concepts named after Niels Henrik Abel
Niels Henrik Abel
Niels Henrik Abel was a Norwegian mathematician who proved the impossibility of solving the quintic equation in radicals.-Early life:...

:

Group theory

  • Abelian group
    Abelian group
    In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...

    , a group in which the binary operation is commutative
    • Category of abelian groups
      Category of abelian groups
      In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category....

       Ab has abelian groups as objects and group homomorphisms as morphisms
  • Metabelian group
    Metabelian group
    In mathematics, a metabelian group is a group whose commutator subgroup is abelian. Equivalently, a group G is metabelian if and only if there is an abelian normal subgroup A such that the quotient group G/A is abelian....

     is a group where the commutator subgroup is abelian
  • Abelianisation

Galois theory

  • Abelian extension
    Abelian extension
    In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is a cyclic group, we have a cyclic extension. More generally, a Galois extension is called solvable if its Galois group is solvable....

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

Functional analysis

  • Abelian von Neumann algebra
    Abelian von Neumann algebra
    In functional analysis, an abelian von Neumann algebra is a von Neumann algebra of operators on a Hilbert space in which all elements commute.The prototypical example of an abelian von Neumann algebra is...

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

Topology and number theory

  • Abelian variety
    Abelian variety
    In mathematics, particularly in algebraic geometry, complex analysis and number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions...

    , a complex torus that can be embedded into projective space
  • Abelian surface
    Abelian surface
    In mathematics, an abelian surface is 2-dimensional abelian variety.One dimensional complex tori are just elliptic curves and are all algebraic, but Riemann discovered that most complex tori of dimension 2 are not algebraic...

    , 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
    Pre-Abelian category
    In mathematics, specifically in category theory, a pre-abelian category is an additive category that has all kernels and cokernels.Spelled out in more detail, this means that a category C is pre-abelian if:...

    , an additive category that has all kernels and cokernels
  • Abelian category
    Abelian category
    In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototype example of an abelian category is the category of abelian groups, Ab. The theory originated in a tentative...

    , a preabelian category in which every monomorphism is a kernel and every epimorphism is a cokernel


In physics
  • A gauge theory
    Gauge theory
    In physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...

     is abelian or non-abelian depending on whether its symmetry group is commutative or non-commutative, respectively

Other uses

  • Abelians
    Abelians
    Abelians were a Christian sect that sprang up in the 4th century near Hippo Regius in north Africa during the reign of Arcadius. They lived in continence as they affirmed Abel did. They were required to be married but were forbidden to consummate the marriage. Each couple was required to adopt two...

    , 4th-century Christian sect
  • Hovhannes Abelian
    Hovhannes Abelian
    Hovhannes Harutyuni Abelian was an Armenian actor, People's Artist of Armenian SSR .- Biography :Since 1882, he worked in Armenian and Russian theatres of Baku and Tiflis...

    (1865–1936), Armenian actor
The source of this article is wikipedia, the free encyclopedia.  The text of this article is licensed under the GFDL.
 
x
OK