Abelian may refer to:

Mathematics

Group theory

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 **

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 *

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. Subgroups of m ...

, a group where the commutator subgroup is abelian *

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

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 *

Abelian surface In mathematics, an abelian surface is a 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 via the Riemann bi ...

, a two-dimensional abelian variety *

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

, a meromorphic function on an abelian variety *

Abelian integral In mathematics, an abelian integral, named after the Norwegian mathematician Niels Henrik Abel, is an integral in the complex plane of the form :\int_^z R(x,w) \, dx, where R(x,w) is an arbitrary rational function of the two variables x and w, wh ...

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

Other mathematics

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 *

Abelian and Tauberian theorems In mathematics, Abelian and Tauberian theorems are theorems giving conditions for two methods of summing divergent series to give the same result, named after Niels Henrik Abel and Alfred Tauber. The original examples are Abel's theorem showing th ...

, in real analysis, used in the summation of divergent series *

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 *

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

, 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 *

Hovhannes Abelian Hovhannes Harutyuni Abelian (, October 23, 1865, Shamakhi, Baku Governorate, Russian Empire - July 1, 1936, Yerevan, Soviet Armenia) was an Armenian actor, People's Artist of the Armenian SSR (1925). Biography From 1882, he worked in Armen ...

(1865–1936), Armenian actor *

Movses Abelian Movses Abelian is an Armenian diplomat who has served as Under-Secretary-General for General Assembly and Conference Management and Coordinator for Multilingualism in the United Nations Secretariat since 2019.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: # C is preadditive, that is enriche ...

, an additive category that has all kernels and cokernels *

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 * Abelians, a 4th-century Christian sect **

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 {{disambiguation