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 commut ...

, 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 of Ab is ...

(Ab), has abelian groups as objects and group homomorphisms as morphisms * Metabelian group, 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 s ...

Topology and number theory

Abelian variety In mathematics, particularly in algebraic geometry, complex analysis and algebraic 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 func ...

, 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 projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular func ...

, 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, whi ...

, 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 of ab ...

, 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 that ...

, 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. When the Galois group is also cyclic, the extension is also called a cyclic extension. Going in the other direction, a Galois extension is called solvabl ...

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

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 the algebra ''L''∞(''X'', μ) for μ ...

, 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 (is invariant) under local transformations according to certain smooth families of operations (Lie groups) ...

with a commutative symmetry group *

Hovhannes Abelian Hovhannes Harutyuni Abelian ( hy, Հովհաննես Աբելյան, 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). Bio ...

(1865–1936), Armenian actor

Mathematics

Group theory

Topology and number theory

Other mathematics

Other uses

See also