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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 .
(In general the Lie bracket of a connected Lie group is always 0 if and only if the Lie group is abelian.) The Lie algebra of the general linear group GL(n, C) of invertible matrices is the vector space M(n, C) of square matrices with the Lie bracket given by [A, B] = AB − BA.
This article gives a table of some common Lie groups and their associated Lie algebras.. The following are noted: the topological properties of the group (dimension; connectedness; compactness; the nature of the fundamental group; and whether or not they are simply connected) as well as on their algebraic properties (abelian; simple; semisimple).
In particular, the Lie algebra of an abelian Lie group (such as the group under addition or the torus group) is abelian. Every finite-dimensional abelian Lie algebra over a field F {\displaystyle F} is isomorphic to F n {\displaystyle F^{n}} for some n ≥ 0 {\displaystyle n\geq 0} , meaning an n -dimensional vector space with Lie bracket zero.
1. A simple Lie group is a connected Lie group that is not abelian which does not have nontrivial connected normal subgroups. 2. A simple Lie algebra is a Lie algebra that is non abelian and has only two ideals, itself and {}. 3.
Suppose G is a closed subgroup of GL(n;C), and thus a Lie group, by the closed subgroups theorem.Then the Lie algebra of G may be computed as [2] [3] = {(;)}. For example, one can use the criterion to establish the correspondence for classical compact groups (cf. the table in "compact Lie groups" below.)
To qualify as an abelian group, the set and operation, (,), must satisfy four requirements known as the abelian group axioms (some authors include in the axioms some properties that belong to the definition of an operation: namely that the operation is defined for any ordered pair of elements of A, that the result is well-defined, and that the ...
There is an unfortunate conflict between the notations for the alternating groups A n and the groups of Lie type A n (q). Some authors use various different fonts for A n to distinguish them. In particular, in this article we make the distinction by setting the alternating groups A n in Roman font and the Lie-type groups A n (q) in italic.