Editing Lie group (section)

=== The Lie algebra associated with a Lie group ===
{{main|Lie group–Lie algebra correspondence}}
To every Lie group we can associate a Lie algebra whose underlying vector space is the tangent space of the Lie group at the identity element and which completely captures the local structure of the group. Informally we can think of elements of the Lie algebra as elements of the group that are "[[infinitesimal]]ly close" to the identity, and the Lie bracket of the Lie algebra is related to the [[commutator]] of two such infinitesimal elements. Before giving the abstract definition we give a few examples:
* The Lie algebra of the vector space '''R'''<sup>''n''</sup> is just '''R'''<sup>''n''</sup> with the Lie bracket given by <br />&nbsp;&nbsp;&nbsp; [''A'',&nbsp;''B''] = 0. <br /> (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 <br />&nbsp;&nbsp;&nbsp; [''A'',&nbsp;''B''] = ''AB''&nbsp;&minus;&nbsp;''BA''.
* If ''G'' is a closed subgroup of GL(''n'', '''C''') then the Lie algebra of ''G'' can be thought of informally as the matrices ''m'' of M(''n'', '''C''') such that 1&nbsp;+&nbsp;ε''m'' is in ''G'', where ε is an infinitesimal positive number with ε<sup>2</sup>&nbsp;=&nbsp;0 (of course, no such real number ε exists). For example, the orthogonal group O(''n'', '''R''') consists of matrices ''A'' with ''AA''<sup>T</sup>&nbsp;=&nbsp;1, so the Lie algebra consists of the matrices ''m'' with (1&nbsp;+&nbsp;ε''m'')(1&nbsp;+&nbsp;ε''m'')<sup>T</sup>&nbsp;=&nbsp;1, which is equivalent to ''m''&nbsp;+&nbsp;''m''<sup>T</sup>&nbsp;=&nbsp;0 because ε<sup>2</sup>&nbsp;=&nbsp;0.
* The preceding description can be made more rigorous as follows. The Lie algebra of a closed subgroup ''G'' of GL(''n'', '''C'''), may be computed as
: <math>\operatorname{Lie}(G) = \{ X \in M(n;\mathbb{C}) | \operatorname{exp}(tX) \in G \text{ for all } t \text{ in } \mathbb{\mathbb{R}} \},</math><ref>{{harvnb|Helgason|1978|loc=Ch. II, § 2, Proposition 2.7}}</ref>{{sfn|ps=|Hall|2015}} where exp(''tX'') is defined using the [[matrix exponential]]. It can then be shown that the Lie algebra of ''G'' is a real vector space that is closed under the bracket operation, {{tmath|1= [X,Y]=XY-YX }}.<ref>{{harvnb|Hall|2015}} Theorem 3.20</ref>

The concrete definition given above for matrix groups is easy to work with, but has some minor problems: to use it we first need to represent a Lie group as a group of matrices, but not all Lie groups can be represented in this way, and it is not even obvious that the Lie algebra is independent of the representation we use.<ref>But see {{harvnb|Hall|2015}}, Proposition 3.30 and Exercise 8 in Chapter 3</ref> To get around these problems we give 
the general definition of the Lie algebra of a Lie group (in 4 steps):
# Vector fields on any smooth manifold ''M'' can be thought of as [[Derivation (abstract algebra)|derivations]] ''X'' of the ring of smooth functions on the manifold, and therefore form a Lie algebra under the Lie bracket [''X'',&nbsp;''Y'']&nbsp;=&nbsp;''XY''&nbsp;&minus;&nbsp;''YX'', because the [[Lie bracket of vector fields|Lie bracket]] of any two derivations is a derivation.
# If ''G'' is any group acting smoothly on the manifold ''M'', then it acts on the vector fields, and the vector space of vector fields fixed by the group is closed under the Lie bracket and therefore also forms a Lie algebra.
# We apply this construction to the case when the manifold ''M'' is the underlying space of a Lie group&nbsp;''G'', with ''G'' acting on ''G''&nbsp;=&nbsp;''M'' by left translations ''L<sub>g</sub>''(''h'')&nbsp;=&nbsp;''gh''. This shows that the space of left invariant vector fields (vector fields satisfying ''L<sub>g</sub>''<sub>*</sub>''X<sub>h</sub>'' =&nbsp;''X<sub>gh</sub>'' for every ''h'' in ''G'', where ''L<sub>g</sub>''<sub>*</sub> denotes the differential of ''L<sub>g</sub>'') on a Lie group is a Lie algebra under the Lie bracket of vector fields.
# Any tangent vector at the identity of a Lie group can be extended to a left invariant vector field by left translating the tangent vector to other points of the manifold. Specifically, the left invariant extension of an element ''v'' of the tangent space at the identity is the vector field defined by ''v''^<sub>''g''</sub>&nbsp;=&nbsp;''L<sub>g</sub>''<sub>*</sub>''v''. This identifies the [[tangent space]] ''T<sub>e</sub>G'' at the identity with the space of left invariant vector fields, and therefore makes the tangent space at the identity into a Lie algebra, called the Lie algebra of ''G'', usually denoted by a [[Fraktur (typeface sub-classification)|Fraktur]] <math>\mathfrak{g}.</math> Thus the Lie bracket on <math>\mathfrak{g}</math> is given explicitly by [''v'',&nbsp;''w'']&nbsp;=&nbsp;[''v''^,&nbsp;''w''^]<sub>''e''</sub>.

This Lie algebra <math>\mathfrak{g}</math> is finite-dimensional and it has the same dimension as the manifold ''G''. The Lie algebra of ''G'' determines ''G'' up to "local isomorphism", where two Lie groups are called '''locally isomorphic''' if they look the same near the identity element.
Problems about Lie groups are often solved by first solving the corresponding problem for the Lie algebras, and the result for groups then usually follows easily.  
For example, simple Lie groups are usually classified by first classifying the corresponding Lie algebras.

We could also define a Lie algebra structure on ''T<sub>e</sub>'' using right invariant vector fields instead of left invariant vector fields. This leads to the same Lie algebra, because the inverse map on ''G'' can be used to identify left invariant vector fields with right invariant vector fields, and acts as &minus;1 on the tangent space ''T<sub>e</sub>''.

The Lie algebra structure on ''T<sub>e</sub>'' can also be described as follows:
the commutator operation
: (''x'', ''y'') → ''xyx''<sup>&minus;1</sup>''y''<sup>&minus;1</sup>
on ''G'' &times; ''G'' sends (''e'',&nbsp;''e'') to ''e'', so its derivative yields a [[bilinear operator|bilinear operation]] on ''T<sub>e</sub>G''. This bilinear operation is actually the zero map, but the second derivative, under the proper identification of tangent spaces, yields an operation that satisfies the axioms of a [[Lie algebra#Definitions|Lie bracket]], and it is equal to twice the one defined through left-invariant vector fields.