Editing Lie group (section)

=== Exponential map ===
{{Main|Exponential map (Lie theory)}}
{{see also|derivative of the exponential map|normal coordinates}}

The [[exponential map (Lie theory)|exponential map]] from the Lie algebra <math>\mathrm{M}(n;\mathbb C)</math> of the [[general linear group]] <math>\mathrm{GL}(n;\mathbb C)</math> to <math>\mathrm{GL}(n;\mathbb C)</math> is defined by the [[matrix exponential]], given by the usual power series:
: <math>\exp(X) = 1 + X + \frac{X^2}{2!} + \frac{X^3}{3!} + \cdots </math>
for matrices {{tmath|1= X }}. If <math>G</math> is a closed subgroup of {{tmath|1= \mathrm{GL}(n;\mathbb C) }}, then the exponential map takes the Lie algebra of <math>G</math> into {{tmath|1= G }}; thus, we have an exponential map for all matrix groups. Every element of <math>G</math> that is sufficiently close to the identity is the exponential of a matrix in the Lie algebra.<ref>{{harvnb|Hall|2015}} Theorem 3.42</ref>

The definition above is easy to use, but it is not defined for Lie groups that are not matrix groups, and it is not clear that the exponential map of a Lie group does not depend on its representation as a matrix group. We can solve both problems using a more abstract definition of the exponential map that works for all Lie groups, as follows.

For each vector <math>X</math> in the Lie algebra <math>\mathfrak{g}</math> of <math>G</math> (i.e., the tangent space to <math>G</math> at the identity), one proves that there is a unique one-parameter subgroup <math>c:\mathbb R\rightarrow G</math> such that {{tmath|1= c'(0)=X }}. Saying that <math>c</math> is a one-parameter subgroup means simply that <math>c</math> is a smooth map into <math>G</math> and that 
: <math>c(s + t) = c(s) c(t)\ </math>
for all <math>s</math> and {{tmath|1= t }}. The operation on the right hand side is the group multiplication in {{tmath|1= G }}. The formal similarity of this formula with the one valid for the [[exponential function]] justifies the definition
: <math>\exp(X) = c(1) .</math>

This is called the '''exponential map''', and it maps the Lie algebra <math>\mathfrak{g}</math> into the Lie group {{tmath|1= G }}. It provides a [[diffeomorphism]] between a [[neighborhood (topology)|neighborhood]] of&nbsp;0 in <math>\mathfrak{g}</math> and a neighborhood of <math>e</math> in {{tmath|1= G }}. This exponential map is a generalization of the exponential function for real numbers (because <math>\mathbb{R}</math> is the Lie algebra of the Lie group of [[positive real numbers]] with multiplication), for complex numbers (because <math>\mathbb{C}</math> is the Lie algebra of the Lie group of non-zero complex numbers with multiplication) and for [[matrix (math)|matrices]] (because <math>M(n, \mathbb{R})</math> with the regular commutator is the Lie algebra of the Lie group <math>\mathrm{GL}(n, \mathbb{R})</math> of all invertible matrices).

Because the exponential map is surjective on some neighbourhood <math>N</math> of {{tmath|1= e }}, it is common to call elements of the Lie algebra '''infinitesimal generators''' of the group {{tmath|1= G }}. The subgroup of <math>G</math> generated by <math>N</math> is  the identity component of {{tmath|1= G }}.

The exponential map and the Lie algebra determine the ''local group structure'' of every connected Lie group, because of the [[Baker&ndash;Campbell&ndash;Hausdorff formula]]: there exists a neighborhood <math>U</math> of the zero element of {{tmath|1= \mathfrak{g} }}, such that for <math>X,Y\in U</math> we have
: <math> \exp(X)\,\exp(Y) = \exp\left(X + Y + \tfrac{1}{2}[X,Y] + \tfrac{1}{12}[\,[X,Y],Y] - \tfrac{1}{12}[\,[X,Y],X] - \cdots \right),</math>
where the omitted terms are known and involve Lie brackets of four or more elements. In case <math>X</math> and <math>Y</math> commute, this formula reduces to the familiar exponential law {{tmath|1= \exp(X)\exp(Y)=\exp(X+Y) }}.

The exponential map relates Lie group homomorphisms. That is, if <math>\phi : G \to H</math> is a Lie group homomorphism and <math>\phi_* : \mathfrak{g} \to \mathfrak{h}</math> the induced map on the corresponding Lie algebras, then for all <math>x \in \mathfrak g</math> we have
: <math>\phi(\exp(x)) = \exp(\phi_{*}(x)) .</math>
In other words, the following diagram [[commutative diagram|commutes]],<ref>{{cite web|url=http://www.math.sunysb.edu/~vkiritch/MAT552/ProblemSet1.pdf |title=Introduction to Lie groups and algebras : Definitions, examples and problems |date=2006 |publisher=State University of New York at Stony Brook |access-date=2014-10-11 |url-status=dead |archive-url=https://web.archive.org/web/20110928024044/http://www.math.sunysb.edu/~vkiritch/MAT552/ProblemSet1.pdf |archive-date=2011-09-28 }}</ref>
[[File:ExponentialMap-01.png|center]]

(In short, exp is a [[natural transformation]] from the functor Lie to the identity functor on the category of Lie groups.)

The exponential map from the Lie algebra to the Lie group is not always [[Surjective function|onto]], even if the group is connected (though it does map onto the Lie group for connected groups that are either compact or nilpotent). For example, the exponential map of {{nowrap|[[SL2(R)|SL(2, '''R''')]]}} is not surjective. Also, the exponential map is neither surjective nor injective for infinite-dimensional (see below) Lie groups modelled on [[Smooth function#Differentiability classes|''C''<sup>∞</sup>]] [[Fréchet space]], even from arbitrary small neighborhood of&nbsp;0 to corresponding neighborhood of&nbsp;1.