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The correspondence between Lie algebras and Lie groups is used in several ways, including in the classification of Lie groups and the representation theory of Lie groups. For finite-dimensional representations, there is an equivalence of categories between representations of a real Lie algebra and representations of the corresponding simply ...
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).
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.)
That is, there may be a group corresponding to the Lie algebra, but it might not be nice enough to be called a Lie group, or the connection between the group and the Lie algebra might not be nice enough (for example, failure of the exponential map to be onto a neighborhood of the identity). It is the "nice enough" that is not universally defined.
The existence of the exponential map is one of the primary reasons that Lie algebras are a useful tool for studying Lie groups. The ordinary exponential function of mathematical analysis is a special case of the exponential map when G {\displaystyle G} is the multiplicative group of positive real numbers (whose Lie algebra is the additive group ...
The subject is part of differential geometry since Lie groups are differentiable manifolds. Lie groups evolve out of the identity (1) and the tangent vectors to one-parameter subgroups generate the Lie algebra. The structure of a Lie group is implicit in its algebra, and the structure of the Lie algebra is expressed by root systems and root data.
One setting in which the Lie algebra representation is well understood is that of semisimple (or reductive) Lie groups, where the associated Lie algebra representation forms a (g,K)-module. Examples of unitary representations arise in quantum mechanics and quantum field theory, but also in Fourier analysis as shown in the following example.
This is a glossary for the terminology applied in the mathematical theories of Lie groups and Lie algebras. For the topics in the representation theory of Lie groups and Lie algebras, see Glossary of representation theory. Because of the lack of other options, the glossary also includes some generalizations such as quantum group