Abelian may refer to:
Mathematics
Group theory
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Abelian group
In mathematics, 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 the order in which they are written. That is, the group operation is commu ...
, a group in which the binary operation is commutative
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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: indeed, every small abelian category can be embedded in Ab.
Properties
The zero object o ...
(Ab), has abelian groups as objects and group homomorphisms as morphisms
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Metabelian group, a group where the commutator subgroup is abelian
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Abelianisation
In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.
The commutator subgroup is important because it is the smallest normal ...
Topology and number theory
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Abelian variety
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a smooth Algebraic variety#Projective variety, projective algebraic variety that is also an algebraic group, i.e., has a group ...
, a complex torus that can be embedded into projective space
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Abelian surface, a two-dimensional abelian variety
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Abelian function, a meromorphic function on an abelian variety
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Abelian integral, a function related to the indefinite integral of a differential of the first kind
Other mathematics
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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 prototypical example of an abelian category is the category o ...
, in category theory, a preabelian category in which every monomorphism is a kernel and every epimorphism is a cokernel
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Abelian and Tauberian theorems, in real analysis, used in the summation of divergent series
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Abelian extension
In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian group, abelian. When the Galois group is also cyclic group, cyclic, the extension is also called a cyclic extension. Going in the other direction, a Galoi ...
, in Galois theory, a field extension for which the associated Galois group is abelian
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Abelian von Neumann algebra, in functional analysis, a von Neumann algebra of operators on a Hilbert space in which all elements commute
Other uses
* Abelian, in physics, a
gauge theory
In physics, a gauge theory is a type of field theory in which the Lagrangian, and hence the dynamics of the system itself, does not change under local transformations according to certain smooth families of operations (Lie groups). Formally, t ...
with a commutative symmetry group
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Hovhannes Abelian (1865–1936), Armenian actor
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Movses Abelian, Armenian diplomat
See also
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Pre-abelian category, an additive category that has all kernels and cokernels
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Niels Henrik Abel
Niels Henrik Abel ( , ; 5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering contributions in a variety of fields. His most famous single result is the first complete proof demonstrating the impossibility of solvin ...
(1802–1829), Norwegian mathematician who gave his name to several different mathematical concepts
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Abelians, a 4th-century Christian sect
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Abel
Abel ( ''Hébel'', in pausa ''Hā́ḇel''; ''Hábel''; , ''Hābēl'') is a biblical figure in the Book of Genesis within the Abrahamic religions. Born as the second son of Adam and Eve, the first two humans created by God in Judaism, God, he ...
, a Biblical figure in the Book of Genesis
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