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Note that a "complex Lie group" is defined as a complex analytic manifold that is also a group whose multiplication and inversion are each given by a holomorphic map. The dimensions in the table below are dimensions over C. Note that every complex Lie group/algebra can also be viewed as a real Lie group/algebra of twice the dimension.
Then a Lie group is defined as a topological group that (1) is locally isomorphic near the identities to an immersely linear Lie group and (2) has at most countably many connected components. Showing the topological definition is equivalent to the usual one is technical (and the beginning readers should skip the following) but is done roughly ...
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.
The following table lists some Lie groups with simple Lie algebras of small dimension. The groups on a given line all have the same Lie algebra. In the dimension 1 case, the groups are abelian and not simple.
In the theory of Lie groups, the exponential map is a map from the Lie algebra of a Lie group to the group, which allows one to recapture the local group structure from the Lie algebra. The existence of the exponential map is one of the primary reasons that Lie algebras are a useful tool for studying Lie groups.
See Table of Lie groups for a list. General linear group, special linear group. SL 2 (R) SL 2 (C) Unitary group, special unitary group. SU(2) SU(3) Orthogonal group, special orthogonal group. Rotation group SO(3) SO(8) Generalized orthogonal group, generalized special orthogonal group. The special unitary group SU(1,1) is the unit sphere in the ...
If G is connected with Lie algebra 𝖌, then its universal covering group G is simply connected. Let G C be the simply connected complex Lie group with Lie algebra 𝖌 C = 𝖌 ⊗ C, let Φ: G → G C be the natural homomorphism (the unique morphism such that Φ *: 𝖌 ↪ 𝖌 ⊗ C is the canonical inclusion) and suppose π: G → G is the universal covering map, so that ker π is the ...
The groups (,), (,), (,) are indeed simple Lie groups, and their finite-dimensional representations coincide [1] with those of their maximal compact subgroups, respectively (), (), (). In the classification of simple Lie algebras , the corresponding algebras are