Download LIE GROUPS AND LIE ALGEBRAS – A FIRST VIEW 1. Motivation

LIE GROUPS AND LIE ALGEBRAS – A FIRST VIEW
ANKE D. POHL
Abstract. Talk in the IRTG Student Seminar 04.04.06. This is a draft
version and thought for internal use only. This exposition is based on
the book “Lie groups” by J.J. Duistermaat and J.A.C. Kolk.
1. Motivation and definitions
The origin of Lie theory is Lie’s idea to develop a “Galois theory” for differential equations and Klein’s “Erlangen Program” of understanding symmetry
groups as the fundamental object to describe a geometry. For example,
symmetric spaces can be completely described by Lie groups.
Definition 1.1. A Lie group is a group G that at the same time is a finitedimensional (real) smooth manifold, such that the two group operations of
G
µ : G × G → G, (x, y) 7→ xy
−1
ι : G → G, x 7→ x
(multiplication),
(inversion),
are smooth mappings.
Definition 1.2. A real Lie algebra is a vector space g over R together with
a bilinear mapping
g × g → g, (X, Y ) 7→ [X, Y ],
called the Lie bracket of g, satisfying
[X, Y ] = −[Y, X]
[X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y ]] = 0
(antisymmetry)
(Jacobi identity)
for all X, Y, Z ∈ g.
The group structure of a Lie group G can be completely recovered from the
algebraic information contained in the Lie bracket of its Lie algebra g, at
least locally near the identity element 1 ∈ G. This is very surprising since
the Lie bracket of g is defined in terms of second-order derivatives at 1.
This fact motivates the study of Lie groups by studying their Lie algebras as
a problem in (multi-) linear algebra because in principle all local information
is known.
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A. POHL
In view of symmetric spaces, the correspondence of the local group structure
of G at 1 and the algebraic structure of g gives to the opportunity to describe
geometric objects or properties purely algebraically with the help of g, e.g.
geodesics have a very simple characterization in terms of certain elements
of g.
2. From a Lie group to its Lie algebra
Let G be a Lie group. The identity element of G will be called 1 or 1G . We
denote the tangent space of the Lie group G at the identity element 1 by
g := T1 G = {D : C ∞ (1) → R | D derivation}.
∼ T1 G × T1 G = g × g we obtain
With the isomorphism T1,1 (G × G) =
T1,1 µ : g × g → g, (X, Y ) 7→ X + Y.
Further we find
T1 ι : g → g, X 7→ −X.
These two mappings give an additive group structure of g and (g, +) becomes
a commutative Lie group. More precisely, using the scalar multiplication of
derivations of C ∞ (1) turns g into a vector space of the same dimension as
G.
For each x ∈ G we define
Ad x : G → G, y 7→ xyx−1 ,
the conjugation by x in G. Because (Ad x)(1) = 1, the tangent mapping of
Ad x at 1 is a linear mapping
Ad x := T1 (Ad x) : g → g,
called the adjoint mapping of x or the infinitesimal conjugation by x in g.
The mapping
Ad : G → GL(g), x 7→ Ad x
is a homomorphism of groups. This mapping is called the adjoint representation of G in g. Due to our definition of Lie groups this mapping is smooth.
If we identify as usual the tangent space at 1 of GL(g) with End(g), then
we obtain the linear mapping
ad := T1 Ad : g → End(g).
For each X, Y ∈ g we put
[X, Y ] := (ad X)(Y ) ∈ g.
Since ad is a linear mapping from g into End(g), the mapping
(X, Y ) 7→ [X, Y ]
is bilinear. With [·, ·] as product, g becomes an algebra over R.
LIE GROUPS AND LIE ALGEBRAS
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Definition 2.1. The tangent space g = T1 G, endowed with the mapping
g × g → g, (X, Y ) 7→ [X, Y ] = (ad X)(Y ) as the product structure, is called
the Lie algebra of the Lie group G. The product [X, Y ] is called the Lie
bracket of X, Y ∈ g.
After the following example we will see that the Lie algebra of a Lie group
is in fact a Lie algebra in the sense of Definition 1.2.
Example 2.2. Let V be a finite-dimensional real vector space. The general
linear group G = GL(V ) is a Lie group (mainly due to Cramer’s rule) and
we get
g = T1 GL(V ) = End(V ).
The adjoint mapping of x ∈ GL(V ) is
Ad x : Y 7→ x ◦ Y ◦ x−1
and as the Lie bracket we get
[X, Y ] = (ad X)(Y ) = X ◦ Y − Y ◦ X,
which is the commutator in End(V ).
Theorem 2.3. Let G be a Lie group and g its Lie algebra. Then for all
X, Y, Z ∈ g we have
[X, Y ] = −[Y, X]
and
[[X, Y ], Z] = [X, [Y, Z]] − [Y, [X, Z]].
The proof extensively use the following proposition which is of its own interest since it tells that a smooth homomorphism of Lie groups comes down
to a smooth homomorphism of their Lie algebras.
Proposition 2.4. Let G and G0 be Lie groups with Lie algebras g and g0 ,
respectively, and suppose that Ψ : G → G0 is a group homomorphism that is
differentiable at 1 ∈ G. Let ψ := T1 Ψ. Then
(a) ψ ◦ Ad x = Ad Ψ(x) ◦ ψ for each x ∈ G.
(b) ψ : g → g0 is a homomorphism of Lie algebras, i.e. it is a linear mapping
g → g0 and in addition
ψ[X, Y ]g = [ψX, ψY ]g0
for all X, Y ∈ g.
Now we see that the Lie algebra of a Lie group is really a Lie algebra.
Clearly, historically the notion of “Lie algebra” derives from the properties
of the Lie algebra of a Lie group.
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A. POHL
3. From the Lie algebra of a Lie group to a Lie subgroup
For each x ∈ G define the left multiplication L(x) : G → G by
L(x) : y 7→ xy.
A vectorfield v on G is called left invariant, if
vL(x)y = Ty L(x)vy
for all y ∈ G.
A left invariant vector field is uniquely determined by its value X := v1 ∈ g
at y = 1:
(1)
vx = T1 L(x)X.
Conversely, since L(xy) = L(x) ◦ L(y) each tangent vector X ∈ g defines a
left invariant vector field on G by (1): We have vy := T1 L(y)X and
vL(x)y = T1 L(xy)X = T1 (L(x) ◦ L(y))X
= Ty L(x) ◦ T1 L(y)X
= Ty L(x)vy .
This left invariant vector field we denote by X L .
Theorem 3.1. For every X ∈ g there is a unique homomorphism h =
hX : (R, +) → (G, ·) that is differentiable at t = 0 and satisfies dh
dt (0) = X.
L
It is equal to the solution curve of X starting at the identity element of G.
The flow of X L is global.
Definition 3.2. For each X ∈ g we define
exp(X) := Φ1 ,
where Φt is the flow of the vector field X L . The mapping
exp : g → G, X 7→ exp X
is called the exponential mapping from g into G.
Proposition 3.3. There is an open neighborhood U of 0 in g and V of 1 in
G such that the exponential mapping is a diffeomorphism from U onto V .
Theorem 3.4. There is an open neighborhood W of (0, 0) in g × g and
an smooth mapping µ : W → g which can be completely described by the
algebraic structure of g such that
exp(X) exp(Y ) = exp µ(X, Y )
for all (X, Y ) ∈ W .
The mapping µ is given by the Baker-Campbell-Hausdorff formula (also
known as Dynkin’s formula or the Taylor expansion of the product at the
LIE GROUPS AND LIE ALGEBRAS
origin in logarithmic coordinates):
∞
X
(−1)k
µ(X, Y ) = Y + X +
k+1
k=1
X
li ,mi ≥0,
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1
l1 + · · · + lk + 1
li +mi >0
(ad X)l1 (ad Y )m1
(ad X)lk (ad Y )mk
(X).
◦
◦ ··· ◦
◦
l1 !
m1 !
lk !
mk !
To get an better idea of this formula we take a look at the expansion up to
order 4:
1
1
1
µ(X, Y ) = X + Y + [X, Y ] + [X, [X, Y ]] + [Y, [Y, X]]
2
12
12
1
5
+ [Y, [X, [Y, X]]] + O(|(X, Y )| )
as (X, Y ) → (0, 0).
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