Lie group

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

•

a

finite-dimensional

,

real-analytic manifold

G

, and

•

two

analytic

maps, one for

multiplication

G

×

G

→

G

and one for

inversion

G

→

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
real-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

×

G

→

G

(

x

,

y

)

↦

x

-

1

⁢

y

∀

x

,

y

∈

G

is analytic.

Next, we describe a natural construction that associates a certain
Lie algebra

𝔤

to every Lie group

G

. Let

e

∈

G

denote the

identity element

of

G

.
For

g

∈

G

let

λ

g

:

G

→

G

denote the

diffeomorphisms

corresponding to left multiplication by

g

.

Definition 2

A

vector field

V

on

G

is called left-invariant if

V

is

invariant

with respect to all left multiplications. To be more
precise,

V

is left-invariant if and only if

(

λ

g

)

*

⁢

(

V

)

=

V

(see push-forward of a vector-field) for all

g

∈

G

.

Proposition 3

The

Lie bracket

of two left-invariant vector fields is
again, a left-invariant vector field.

Proof.

Let

V

1

,

V

2

be left-invariant vector fields, and let

g

∈

G

.
The bracket

operation

is covariant with respect to diffeomorphism, and
in particular

(

λ

g

)

*

⁢

[

V

1

,

V

2

]

=

[

(

λ

g

)

*

⁢

V

1

,

(

λ

g

)

*

⁢

V

2

]

=

[

V

1

,

V

2

]

.

Q.E.D.

Definition 4

The Lie algebra of

G

, denoted hereafter by

g

, is the

vector space

of all left-invariant vector fields equipped with the vector-field
bracket.

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

infinitesimal

right multiplication.

Proposition 5

Let

a

∈

T

e

⁢

G

and let

V

be a left-invariant vector-field such that

V

e

=

a

. Then for all

g

∈

G

we have

V

g

=

(

λ

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

γ

:

(

-

ϵ

,

ϵ

)

→

G

passing through the identity element with

velocity

a

; i.e.

γ

⁢

(

0

)

=

e

,

γ

′

⁢

(

0

)

=

a

.

The above

proposition

is then
saying that the curve

t

↦

g

⁢

γ

⁢

(

t

)

,

t

∈

(

-

ϵ

,

ϵ

)

passes through

g

at

t

=

0

with velocity

V

g

.

Thus we see that a left-invariant vector field is completely
determined by the value it takes at

e

, and that therefore

𝔤

is

isomorphic

, as a vector space to

T

e

⁢

G

.

Of course, we can also consider the Lie algebra of right-invariant
vector fields. The resulting Lie-algebra is anti-isomorphic (the
order in the bracket is reversed) to the Lie algebra of left-invariant
vector fields. Now it is a general principle that the

group inverse

operation gives an anti-isomorphism between left and right

group
actions

. So, as one may well expect, the anti-isomorphism between the
Lie algebras of left and right-invariant 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

∞

or even

C

k

,

k

=

1

,

2

,

…

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 real-analytic structure.

Indeed, one can go even further and show that even

C

0

suffices. In
other words, a topological group that is also a finite-dimensional

topological manifold

possesses a compatible analytic structure. This
result was formulated by Hilbert as his

http://www.reed.edu/ wieting/essays/LieHilbert.pdf

fifth 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 complex-analytic. The theory of
complex Lie groups requires the notion of a holomorphic vector-field.
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

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

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