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 ofdata

• a finite-dimensional, real-analytic manifold $G$, and

• two analytic maps, one for multiplication $G\times G\to G$ and one for inversion $G\to G$, which obeythe appropriategroup axioms.

Thus, a homomorphism in the category of Lie groups is a grouphomomorphism that is simultaneously an analytic mapping between tworeal-analytic manifolds.

One can equivalently define a Lie group $G$ using the following easy characterization

Proposition 1

A finite-dimensional 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 certainLie algebra $𝔤$ to every Lie group $G$. Let $e\in G$ denote theidentity element of $G$.For $g\in G$ let ${\lambda }_g:G\to G$ denote thediffeomorphisms corresponding to left multiplication by $g$.

Definition 2

A vector field $V$ on $G$ is called left-invariant if $V$ isinvariant with respect to all left multiplications. To be moreprecise, $V$ is left-invariant if and only if

$${({\lambda }_g)}_{*}(V)=V$$

(see push-forward of a vector-field) for all$g\mathrm{\in }G$.

Proposition 3

The Lie bracket of two left-invariant vector fields isagain, a left-invariant vector field.

Proof.Let $V_1,V_2$ be left-invariant vector fields, and let $g\in G$.The bracket operation is covariant with respect to diffeomorphism, andin 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 spaceof all left-invariant vector fields equipped with the vector-fieldbracket.

Now a right multiplication is invariant with respect to allleft multiplications, and it turns out that we can characterize aleft-invariant vector field as being an infinitesimal right multiplication.

Proposition 5

Let $a\mathrm{\in }{T}_e\mathit{}G$ and let $V$ be a left-invariant vector-field 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

$$\gamma :(-ϵ,ϵ)\to G$$

passing through the identity element with velocity $a$; i.e.

$$\gamma (0)=e,{\gamma }^{\prime }(0)=a.$$

The above proposition is thensaying that the curve

$$t\mapsto g\gamma (t),t\in (-ϵ,ϵ)$$

passes through$g$ at $t=0$ with velocity $V_g$.

Thus we see that a left-invariant vector field is completelydetermined by the value it takes at $e$, and that therefore $𝔤$ isisomorphic, as a vector space to $T_{e}G$.

Of course, we can also consider the Lie algebra of right-invariantvector fields. The resulting Lie-algebra is anti-isomorphic (theorder in the bracket is reversed) to the Lie algebra of left-invariantvector fields. Now it is a general principle that the group inverseoperation gives an anti-isomorphism between left and right groupactions. So, as one may well expect, the anti-isomorphism between theLie algebras of left and right-invariant vector fields can be realizedby considering the linear action of the inverse operation on $T_{e}G$.

Finally, let us remark that one can induce the Lie algebra structuredirectly 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 endowsthis manifold with a compatible real-analytic structure.

   Indeed, one can go even further and show that even $C^0$ suffices. Inother words, a topological group that is also a finite-dimensionaltopological manifold possesses a compatible analytic structure. Thisresult was formulated by Hilbert as his http://www.reed.edu/ wieting/essays/LieHilbert.pdffifth problem, andproved in the 50’s by Montgomery and Zippin.

2. One can also speak of a complex Lie group, in which case $G$ and themultiplication mapping are both complex-analytic. The theory ofcomplex Lie groups requires the notion of a holomorphic vector-field.Not withstanding this complication, most of the essential features ofthe real theory carry over to the complex case.

3. The name “Lie group” honours the Norwegian mathematicianSophus Lie who pioneered and developed the theory of continuoustransformation groups and the corresponding theory of Lie algebrasof vector fields (the group’s infinitesimal generators, as Lietermed them). Lie’s original impetus was the study of continuoussymmetry of geometric objects and differential equations.

   The scope of the theory has grown enormously in the 100+ years ofits existence. The contributions of Elie Cartan and ClaudeChevalley figure prominently in this evolution. Cartan isresponsible for the celebrated ADE classification of simple Liealgebras, as well as for charting the essential role played by Liegroups in differential geometry and mathematical physics. Chevalleymade key foundational contributions to the analytic theory, and didmuch to pioneer the related theory of algebraic groups. ArmandBorel’s book “Essays in the History of Lie groups and algebraicgroups” is the definitive source on the evolution of the Lie groupconcept. Sophus Lie’s contributions are the subject of a number ofexcellent articles by T. Hawkins.

Title	Lie group
Canonical name	LieGroup
Date of creation	2013-05-19 19:12:53
Last modified on	2013-05-19 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	left-invariant
Defines	right-invariant