Lie Groups; Action on a Manifold

Chapter I: Sheaves and Differential Manifolds:

Definitions and Examples

4 Lie Groups; Action on a Manifold

4.1. Denition. A Lie group G consists of two structures on the same set G, namely

it is a dierential manifold and has also a group structure. The two structures are

interrelated by the assumptions that the group law G × G → G and the group inverse

G → G are both dierentiable.

4.2. Examples.

1) Any countable group with the discrete topology is a Lie group in our sense. (Count-

ability is required because in our denition, manifolds are supposed to have a

countable base of open sets.)

2) The real line R is a Lie group under addition since the maps R ×R → R given by

(x, y) 7→ x + y and R → R given by x 7→ −x are dierentiable.

3) It is also clear that R

n

is a Lie group under addition.

4) The multiplicative group C

×

consisting of nonzero complex numbers is an open

submanifold of C and is actually a Lie group under multiplication.

5) The groups GL(n, R) or GL(n, C), which are open submanifolds of R

n

2

and C

n

2

,

are Lie groups. Indeed, the group composition is the restriction of a polynomial

map R

n

2

× R

n

2

→ R

n

2

. If V is any vector space of nite-dimension over R or C,

then the group GL(V ) of linear automorphisms is a Lie group.

6) The orthogonal group O(n, R) (resp. the unitary group U(n)) is a Lie group under

matrix multiplication.

4.3. Exercise. The quotient of GL(V ) by its centre, namely nonzero scalar matrices

(or automorphisms), is called the projective linear group P GL(V ). Show that it is a Lie

group.

4.4. Denition. A homomorphism G → H of Lie groups is a group homomor-

phism which is also dierentiable. A homomorphism of G into GL(V ) is said to be a

representation of G in the vector space V .

It is clear that the composite of a homomorphism G

1

→ G

2

of Lie groups and another

from G

2

→ G

3

is a homomorphism from G

1

→ G

3

.

4.5. Denition. A Lie subgroup H of a Lie group G is a Lie group H with an

injective homomorphism of H into G.

4.6. Remark. For any g ∈ G, the (right) translation map G → G given by x 7→ xg

is of course dierentiable, being the composite of the inclusion G → G × G given by

30

Section 4: Lie Groups; Action on a Manifold

g 7→ (x, g) and the group operation. Its inverse is translation by g

−1

. Hence right (and

similarly left) translations are dieomorphisms.

4.7. Denition. Let G be a Lie group and M a dierential manifold. An action

of G on M is a dierentiable map G × M → M denoted (g, m) 7→ gm such that

g

1

(g

2

m) = (g

1

g

2

)(m) for all g

1

, g

2

∈ G and m ∈ M and 1.m = m for all m ∈ M .

In Physics, the role of the Lie group is that of the symmetries of the system. Often

the most important physical insight turns out to be the intuition for the appropriate

group of symmetries.

4.8. Examples.

1) The proper orthogonal group SO(3) acting on R

3

and the group generated by it

and translations (called the Euclidean motion group) are the group of symmetries

in the study of motion of rigid bodies.

2) The group of all linear transformations of R

4

which leave the symmetric bilinear

form

((x

1

, x

2

, x

3

, x

4

), (y

1

, y

2

, y

3

, y

4

)) 7→ −x

1

y

1

+ x

2

y

2

+ x

3

y

3

+ x

4

y

4

invariant, is called the homogeneous Lorentz group. If we consider the group

generated by this group and translations, then it is called the inhomogeneous

Lorentz group. Both these groups act on the dierential manifold R

4

. This is the

symmetry group for the theory of special relativity.

In quantum physics, one is often interested in representations of G in the projective

unitary group (namely the group of unitary operators, modulo scalars) of a Hilbert

space, but we will not deal with them in this book.

Exercises

1) Which of the following are sheaves on R

n

:

a) For every open set dene F(U ) to be the space of square integrable functions

on U.

b) F(U) is the set of Lebesgue measurable functions on U .

c) F(U) consists of continuous functions on U which are restrictions of contin-

uous functions on R

n

.

2) Show that the stalk at 0 of the sheaf of dierentiable functions on R

n

, n ≥ 1, is

an innite-dimensional vector space over R.

3) Show that the étale space associated to the sheaf of dierentiable functions on R

is not Hausdor.

31

Chapter I: Sheaves and Differential Manifolds:

Definitions and Examples

4) Show that any section of the sheaf A of continuous functions on a closed set of a

normal topological space X can be extended to a section over the whole of X.

5) Determine for what values of a

i

and c is the intersection of the hyperplane

∑

a

i

x

i

=

c with the sphere

∑

x

2

i

= 1, a closed submanifold of R

n

.

6) Let M be a dierential manifold and f a dierentiable function on it. Realise the

open submanifold of M given by f 6= 0 as a closed submanifold of M × R.

7) Consider the map x 7→ x

2

of GL(2, R) (resp. GL(2, C)) into itself and nd its

image. Is the image a submanifold?

8) Show that the space of nonzero nilpotent (2, 2) matrices is a closed submanifold

of the space of nonzero matrices.

9) If G

1

, G

2

are Lie groups, show that the product manifold G

1

×G

2

with the direct

product structure is also a Lie group.

10) Interpret the Jordan canonical form for matrices, as describing the orbits under

the action of GL(2, C) on the space M (2, C) of all matrices given by g.A = gAg

−1

.

Determine which orbits are closed submanifolds.

32