In geometry, an abelian Lie group is a Lie group that is an abelian group.
A connected abelian real Lie group is isomorphic to .[1] In particular, a connected abelian (real) compact Lie group is a torus; i.e., a Lie group isomorphic to . A connected complex Lie group that is a compact group is abelian and a connected compact complex Lie group is a complex torus; i.e., a quotient of by a lattice.
Let A be a compact abelian Lie group with the identity component . If is a cyclic group, then is topologically cyclic; i.e., has an element that generates a dense subgroup.[2] (In particular, a torus is topologically cyclic.)
In geometry, an abelianLiegroup is a Liegroup that is an abeliangroup. A connected abelian real Liegroup is isomorphic to R k × ( S 1 ) h {\displaystyle...
classification of Liegroups. Analogously to abelian, nilpotent, and solvable groups, one can define abelian, nilpotent, and solvable Lie algebras. A Lie algebra...
mathematics, an abeliangroup, also called a commutative group, is a group in which the result of applying the group operation to two group elements does...
group; and whether or not they are simply connected) as well as on their algebraic properties (abelian; simple; semisimple). For more examples of Lie...
1-dimensional compact, connected Liegroup. Moreover, every n {\displaystyle n} -dimensional compact, connected, abelianLiegroup is isomorphic to T n {\displaystyle...
simple connected Liegroups. Gsol/Gnil is abelian. A connected abelianLiegroup is isomorphic to a product of copies of R and the circle group S1. Gnil/1 is...
In mathematics, specifically in group theory, an elementary abeliangroup is an abeliangroup in which all elements other than the identity have the same...
nilpotent group is a group that is "almost abelian". This idea is motivated by the fact that nilpotent groups are solvable, and for finite nilpotent groups, two...
theory, locally compact abeliangroups are abeliangroups which have a particularly convenient topology on them. For example, the group of integers (equipped...
cyclic group. An abeliangroup, also called a commutative group, is a group in which the result of applying the group operation to two group elements...
example would be any abelianLiegroup. This is because any such group is a nilpotent Liegroup. For example, one can take the group of real numbers under...
Free abeliangroup Free group Free product Generating set of a groupGroup cohomology Group extension Presentation of a group Product of group subsets...
mathematics, specifically in group theory, the phrase group of Lie type usually refers to finite groups that are closely related to the group of rational points...
groups and free Lie algebras. The group (Z,+) of integers is free of rank 1; a generating set is S = {1}. The integers are also a free abeliangroup,...
the classification of abeliangroups: according to the fundamental theorem of finite abeliangroups, every finite abeliangroup can be expressed as the...
families of non-abelian finite simple groups may be considered to be of Lie type. One of 16 families of groups of Lie type The Tits group is generally considered...
The modular group may be realised as a quotient of the special linear group SL(2, Z). If n ≥ 2, then the group GL(n, F) is not abelian. If V is a vector...
specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abeliangroups using extensions. Equivalently...
also an example of a compact abelianLiegroup. This follows from the fact that the unit circle is a compact abelianLiegroup (when identified with the...
In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelianLie algebra without any...