Editing Lie algebra (section)

== Relation to Lie groups ==
{{main|Lie group–Lie algebra correspondence}}
[[Image:Image Tangent-plane.svg|thumb| The tangent space of a [[sphere]] at a point <math>x</math>. If <math>x</math> were the identity element of a Lie group, the tangent space would be a Lie algebra.]]
Although Lie algebras can be studied in their own right, historically they arose as a means to study [[Lie group]]s.

The relationship between Lie groups and Lie algebras can be summarized as follows. Each Lie group determines a Lie algebra over <math>\mathbb{R}</math> (concretely, the tangent space at the identity). Conversely, for every finite-dimensional Lie algebra <math>\mathfrak g</math>, there is a connected Lie group <math>G</math> with Lie algebra <math>\mathfrak g</math>. This is [[Lie's third theorem]]; see the [[Baker–Campbell–Hausdorff formula]]. This Lie group is not determined uniquely; however, any two Lie groups with the same Lie algebra are ''locally isomorphic'', and more strongly, they have the same [[universal cover]]. For instance, the special orthogonal group [[SO(3)]] and the special unitary group [[SU(2)]] have isomorphic Lie algebras, but SU(2) is a [[simply connected]] double cover of SO(3).

For ''simply connected'' Lie groups, there is a complete correspondence: taking the Lie algebra gives an [[equivalence of categories]] from simply connected Lie groups to Lie algebras of finite dimension over <math>\mathbb{R}</math>.<ref>{{harvnb|Varadarajan|1984|loc=Theorems 2.7.5 and 3.15.1.}}</ref>

The correspondence between Lie algebras and Lie groups is used in several ways, including in the [[list of simple Lie groups|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 connected Lie group. This simplifies the representation theory of Lie groups: it is often easier to classify the representations of a Lie algebra, using linear algebra.

Every connected Lie group is isomorphic to its universal cover modulo a [[discrete group|discrete]] central subgroup.<ref>{{harvnb|Varadarajan|1984|loc=section 2.6.}}</ref> So classifying Lie groups becomes simply a matter of counting the discrete subgroups of the [[Center (group theory)|center]], once the Lie algebra is known. For example, the real semisimple Lie algebras were classified by Cartan, and so the classification of semisimple Lie groups is well understood.

For infinite-dimensional Lie algebras, Lie theory works less well. The exponential map need not be a local [[homeomorphism]] (for example, in the diffeomorphism group of the circle, there are diffeomorphisms arbitrarily close to the identity that are not in the image of the exponential map). Moreover, in terms of the existing notions of infinite-dimensional Lie groups, some infinite-dimensional Lie algebras do not come from any group.<ref>{{harvnb|Milnor|2010|loc=Warnings 1.6 and 8.5.}}</ref>

Lie theory also does not work so neatly for infinite-dimensional representations of a finite-dimensional group. Even for the additive group <math>G=\mathbb{R}</math>, an infinite-dimensional representation of <math>G</math> can usually not be differentiated to produce a representation of its Lie algebra on the same space, or vice versa.<ref>{{harvnb|Knapp|2001|loc=section III.3, Problem III.5.}}</ref> The theory of [[Harish-Chandra module]]s is a more subtle relation between infinite-dimensional representations for groups and Lie algebras.