Lie group
A Lie group is a group endowed with a compatible^{} analytic structure (http://planetmath.org/ComplexAnalyticManifold). To be more precise, Lie group structure^{} consists of two kinds of data

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a finitedimensional^{}, realanalytic manifold $G$, and

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two analytic maps, one for multiplication^{} $G\times G\to G$ and one for inversion^{} $G\to G$, which obey the appropriate group axioms.
Thus, a homomorphism^{} in the category of Lie groups is a group homomorphism that is simultaneously an analytic mapping between two realanalytic manifolds.
One can equivalently define a Lie group $G$ using the following easy characterization^{}
Proposition 1
A finitedimensional real analytic manifold $G$ is a Lie algebra iff the map
$$G\times G\to G\mathit{\hspace{1em}\hspace{1em}}(x,y)\mapsto {x}^{1}y\mathit{\hspace{1em}}\forall x,y\in G$$ 
is analytic.
Next, we describe a natural construction that associates a certain Lie algebra $\U0001d524$ to every Lie group $G$. Let $e\in G$ denote the identity element^{} of $G$. For $g\in G$ let ${\lambda}_{g}:G\to G$ denote the diffeomorphisms corresponding to left multiplication by $g$.
Definition 2
A vector field $V$ on $G$ is called leftinvariant if $V$ is invariant^{} with respect to all left multiplications. To be more precise, $V$ is leftinvariant if and only if
$${({\lambda}_{g})}_{*}(V)=V$$ 
(see pushforward of a vectorfield) for all $g\mathrm{\in}G$.
Proposition 3
The Lie bracket of two leftinvariant vector fields is again, a leftinvariant vector field.
Proof. Let ${V}_{1},{V}_{2}$ be leftinvariant vector fields, and let $g\in G$. The bracket operation^{} is covariant with respect to diffeomorphism, and in particular
$${({\lambda}_{g})}_{*}[{V}_{1},{V}_{2}]=[{({\lambda}_{g})}_{*}{V}_{1},{({\lambda}_{g})}_{*}{V}_{2}]=[{V}_{1},{V}_{2}].$$ 
Q.E.D.
Definition 4
The Lie algebra of $G$, denoted hereafter by $\mathrm{g}$, is the vector space^{} of all leftinvariant vector fields equipped with the vectorfield bracket.
Now a right multiplication is invariant with respect to all left multiplications, and it turns out that we can characterize a leftinvariant vector field as being an infinitesimal right multiplication.
Proposition 5
Let $a\mathrm{\in}{T}_{e}\mathit{}G$ and let $V$ be a leftinvariant vectorfield such that ${V}_{e}\mathrm{=}a$. Then for all $g\mathrm{\in}G$ we have
$${V}_{g}={({\lambda}_{g})}_{*}(a).$$ 
The intuition here is that $a$ gives an infinitesimal displacement from the identity element and that ${V}_{g}$ gives a corresponding infinitesimal right displacement away from $g$. Indeed consider a curve
$$\gamma :(\u03f5,\u03f5)\to G$$ 
passing through the identity element with velocity $a$; i.e.
$$\gamma (0)=e,{\gamma}^{\prime}(0)=a.$$ 
The above proposition^{} is then saying that the curve
$$t\mapsto g\gamma (t),t\in (\u03f5,\u03f5)$$ 
passes through $g$ at $t=0$ with velocity ${V}_{g}$.
Thus we see that a leftinvariant vector field is completely determined by the value it takes at $e$, and that therefore $\U0001d524$ is isomorphic, as a vector space to ${T}_{e}G$.
Of course, we can also consider the Lie algebra of rightinvariant vector fields. The resulting Liealgebra is antiisomorphic (the order in the bracket is reversed) to the Lie algebra of leftinvariant vector fields. Now it is a general principle that the group inverse operation gives an antiisomorphism between left and right group actions^{}. So, as one may well expect, the antiisomorphism between the Lie algebras of left and rightinvariant vector fields can be realized by considering the linear action of the inverse^{} operation on ${T}_{e}G$.
Finally, let us remark that one can induce the Lie algebra structure directly on ${T}_{e}G$ by considering adjoint action of $G$ on ${T}_{e}G$.
History and motivation.
Examples.
Notes.

1.
No generality is lost in assuming that a Lie group has analytic, rather than ${C}^{\mathrm{\infty}}$ or even ${C}^{k},k=1,2,\mathrm{\dots}$ structure. Indeed, given a ${C}^{1}$ differential manifold with a ${C}^{1}$ multiplication rule, one can show that the exponential mapping endows this manifold with a compatible realanalytic structure.
Indeed, one can go even further and show that even ${C}^{0}$ suffices. In other words, a topological group that is also a finitedimensional topological manifold possesses a compatible analytic structure. This result was formulated by Hilbert as his http://www.reed.edu/ wieting/essays/LieHilbert.pdffifth problem, and proved in the 50’s by Montgomery and Zippin.

2.
One can also speak of a complex Lie group, in which case $G$ and the multiplication mapping are both complexanalytic. The theory of complex Lie groups requires the notion of a holomorphic vectorfield. Not withstanding this complication, most of the essential features of the real theory carry over to the complex case.

3.
The name “Lie group” honours the Norwegian mathematician Sophus Lie who pioneered and developed the theory of continuous transformation groups and the corresponding theory of Lie algebras of vector fields (the group’s infinitesimal generators, as Lie termed them). Lie’s original impetus was the study of continuous symmetry of geometric objects and differential equations.
The scope of the theory has grown enormously in the 100+ years of its existence. The contributions of Elie Cartan and Claude Chevalley figure prominently in this evolution. Cartan is responsible for the celebrated ADE classification of simple Lie algebras^{}, as well as for charting the essential role played by Lie groups in differential geometry and mathematical physics. Chevalley made key foundational contributions to the analytic theory, and did much to pioneer the related theory of algebraic groups. Armand Borel’s book “Essays in the History of Lie groups and algebraic groups” is the definitive source on the evolution of the Lie group concept^{}. Sophus Lie’s contributions are the subject of a number of excellent articles by T. Hawkins.
Title  Lie group 
Canonical name  LieGroup 
Date of creation  20130519 19:12:53 
Last modified on  20130519 19:12:53 
Owner  rmilson (146) 
Last modified by  jocaps (12118) 
Numerical id  21 
Author  rmilson (12118) 
Entry type  Definition 
Classification  msc 22E10 
Classification  msc 22E15 
Related topic  Group 
Related topic  LieAlgebra 
Related topic  SimpleAndSemiSimpleLieAlgebras2 
Defines  leftinvariant 
Defines  rightinvariant 