Download Notes - Brown math department

2. Lie groups: basic definitions
2.1 Reminders from differential geometry
Notation
In this text, all manifolds are C ∞ manifolds and ”smooth” means C ∞ . Denote by Tm M the tangent
space to M at m and denote by T M = ∪m∈M Tm M (disjoint unions of tangent spaces) the tangent
bundle. Denote by Vect(M) the space of vector fields on M. Fr a morphism f : X → Y and for
x ∈ X, denote f∗ : Tx X → Ty Y the corresponding map of tangent spaces.
Definition 1. A morphism f : X → Y is called an immersion if rankf∗ =dim X for all x ∈ X. For
an immersion f, we can choose local coordinates in a neighborhood of x and in a neighborhood of
f (x) such that f is represented by f (x1 , · · · , xn ) = (x1 , · · · , xn , 0, · · · , 0) (Local immersion theorem).
Definition 2. An immersed submanifold N in a manifold M is a subset with a structure of a
manifold such that inclusion map i : N ,→ M is an immersion(i.e., not a topological embedding nor
homeomorphism). Note that the manifold structure of N is not unique. Note that Tp N ⊂ Tp M as
a subspace.
Definition 3. An embedded submanifold N ⊂ M is an immersed submanifold such that the inclusion
map i : N ,→ M is a homeomorphism: the smooth structure on N is uniquely determined by the
smooth structure of M.
Denote ”submanifold” for embedded submanifolds.
2.2 Lie groups, subgroups, and cosets
Definition 4. A Lie group is a set G where G is a group and G is a manifold(In this definition, simply
manifold, not an embedded submanifold). Multiplication and inversion map are smooth maps. A
morphism f of Lie groups is a smooth map preserving the group operation, that is, f(gh)=f(g)f(h),
f(1)=1.
Note that complex Lie groups are defined by replacing ”smooth” to ”analytic” of the above definition and ”submanifold” will mean ”complex analytic submanifold”. In C 0 , C 1 , andC 2 respectively.
Example 5. Examples of Lie groups
(1) Rn with addition, R∗ with multiplication.
(2) GL(n, R), GL(n, C)
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Note that GL(n, R) ⊂ Rn and {0} is closed. Thus R\{0} is open and the inverse image
of the continuous map, i.e., determinant, is also open. Therefore GL(n, R) is a manifold.
Therefore, it is a Lie group.
(3) SU (2) = {A ∈ GL(2, C)|AĀt = 1, detA = 1}
Note that
α β
2
2
SU(2) =
: α, β ∈ C, |α| + |β| = 1
−β̄ ᾱ
Letting α = x1 + ix2 , β = x3 + ix4 , xi ∈ R, SU (2) ' S 3 ⊂ R4 .(3-dimensional manifold)
Thus SU(2) is a Lie group.
Note that any finite group is a 0-dimensional Lie group. In order to separately think finite part
we need the following theorem.
Theorem 6. Let G be a Lie group. Denote by G0 the connected component of identity. Then G0
is a normal subgroup of G and is a Lie group and the quotient group G/G0 is discrete.
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This theorem reduces the study of Lie groups to the study of finite groups and connected Lie
groups.
Theorem 7. If G is a connected Lie group, then its universal cover G̃ has a canonical structure of
a Lie group such that the covering map p : G̃ → G is a morphism and ker(p) = π1 (G) and Kerp is
a discrete central subgroup (i.e., a subgroup which lies in the center of the group).
Proof. If M, N are connected manifolds, then any continuous map f : M → N can be lifted to the
map of universal covers f˜ : M̃ → Ñ . Moreover, we can uniquely lift the map fixing one point (m,
f(m)) and its lifting. Using the theorem on the inversion map i : G → G and on the multiplication
map G × G → G, we can get G̃ is a lie group. To see Kerp = π1 (G), recall the definition of covering
space G̃ = {[γ] : γ : I → G, γ(0) = 1G } and the covering map p : [γ] 7→ γ(1). Then Kerp consists
of the equivalence classes with γ(0) = γ(1) = 1G , i.e., Kerp = π1 (G)(Since G is connected the
fundamental group of G does not depend on the choice of the base point).
Definition 8. A closed Lie subgroup H of a Lie group G is a subgroup which is also a submanifold
(i.e., embedded submanifold).
Theorem 9. (1) Any closed Lie subgroup is closed in G.
(2) Any closed subgroup of a Lie group is a closed Lie subgroup.
Corollary 10. (1) If G is a connected Lie group and U is a neighborhood of 1, then U generates G.
(2) Let f : G1 → G2 be a morphism of Lie groups with G2 connected, such that f∗ : T1 G1 → T1 G2
is surjective. Then f is surjective.
Proof. Let H be the subgroup generated by U. For any element h ∈ H, hU̇ is a neighborhood of h in
H. Thus H is open in G, so H is a submanifold, i.e., H is a closed Lie group. By the above theorem,
H is closed, and is nonempty, H = G (Since G is connected, only clopen sets are the empty set and
G itself).
Given a closed Lie subgroup H ⊂ G, define cosets and the coset space G/H. The following
theorem indicates that the coset space is a manifold.
Theorem 11. H ⊂ G
(1) Let G be a Lie group of dimension n and H ⊂ G a closed Lie subgroup of dimension k. Then
the coset space G/H has a natural structure of a manifold of dimension n − k such that the
canonical map p : G → G/H is a fiber bundle (i.e., locally product space but globally may have
other topological structures), with fiber diffeomorphic to H. The tangent space at 1̄ = p(1) is
T1̄ (G/H) = T1 G/T1 H(Quotient vector space which is well-defined by the observation before).
(2) If H is a normal closed Lie subgroup then G/H has a canonical structure of a Lie group.
Corollary 12. Let H be a closed Lie subgroup of a Lie group G.
(1) If H is connected, then the set of connected components π0 (G) = π0 (G/H). In particular, if
H and G/H are connected, then so is G.
(2) If G and H are connected, then there is an exact sequence of fundamental groups
π2 (G/H) → π1 (H) → π1 (G) → π1 (G/H) → {1}.
We will use the exact sequence later to compute fundamental groups of classical groups such as
GL(n, K)
2.3. Lie subgroups and homomorphism theorem
The following example shows that the image of a morphism may not be a closed Lie subgroup.
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Example 13. Let G1 = R and G2 = T 2 = R2 /Z2 . Define the map
f : G1 → G2 ,
f (t) = (t mod Z, αt mod Z)
,where α is some fixed irrational number. Then the image of this map is dense in T 2 (irrational
winding on the torus). However, the image is not equal to T 2 and therefore the image is not closed
in T 2 . Thus the image of closed Lie subgroup is may not be a closed Lie subgroup.
Therefore, the definition of closed Lie subgroup is too restrictive. Introduce a more general notion
of a subgroup!
Definition 14. An Lie subgroup in a Lie group H ⊂ G is an immersed submanifold which is also
a subgroup.
In the definition of closed Lie subgroup, we required the submanifold to be an embedded manifold(i.e., require homeomorphism, topological embedding) but we only require it to be an immersed
submanifold for an Lie subgroup. In this situation, H is itself a Lie group and i : H ,→ G is a
morphism of Lie groups. Note that every closed Lie subgroup is a Lie subgroup, yet the converse is
false (the above example). Also note that if a Lie subgroup is closed in G, then it is a closed Lie
subgroup (the last statement requires proof).
Theorem 15. Let f : G1 → G2 be a morphism of Lie groups. Then H = Kerf is a normal closed
Lie subgroup in G1 , and f induces an injective morphism G1 /H → G2 , which is an immersion.
Thus, Imf is a Lie subgroup in G2 . If Imf is an (embedded) submanifold, then it is a closed Lie
subgroup in G2 and f gives an isomorphism of Lie groups G1 /H ' Imf .
The proof would be discussed later.