Universal Cover

Connectedness

A matrix Lie group G is said to be path-connected (in topology) if given any two matrices A and B in G, there exists a continuous path A(t), a ≤ t ≤ b, lying in G with A(a) = A and A(b) = B. A matrix Lie group is connected if and only if it is path-connected. 

A matrix Lie group G which is not connected can be decomposed (uniquely) as a union of several pieces, called components, such that two elements of the same component can be joined by a continuous path, but two elements of different components cannot.

Examples:
Group      Connected   Components
GL(n;C)      yes             1
SL(n;C)      yes             1
GL(n;R)      no              2
SL(n;R)      yes             1
O(n)           no               2
SO(n)         yes             1
U(n)           yes             1
SU(n)         yes             1
O(n;1)        no               4
SO(n;1)      no               2
Heisenberg yes            1
E(n)           no               2
P(n;1)        no               4


Simple Connectedness

A matrix Lie group G is said to be simply connected if it is connected and, in addition, every loop in G can be shrunk continuously to a point in G. More precisely, assume that G is connected. Then, G is simply connected if given any continuous path A(t), 0 ≤ t ≤ 1, lying in G with A(0) = A(1), there exists a continuous function A(s,t), 0 ≤ s,t ≤ 1, taking values in G and having the following properties:

(1) A(s,0) = A(s,1) for all s, is a loop condition;
(2) A(0,t) = A(t), when s=0 the loop is the specified loop A(t);
(3) A(1,t) = A(1,0) for all t, when s=1 the loop is a point.


Group versus Lie Algebra Homomorphisms

(A) Lie algebra homomorphism

Every Lie group homomorphism give rise to a Lie algebra homomorphism. Let G and H be matrix Lie groups, with Lie algebras g and h, respectively. Suppose that Φ : G --> H is a Lie group homomorphism. Then, there exists a unique real linear map ϕ :   g --> h such that  for all X ∈ g. The map ϕ has the following additional properties:

(1) ϕ (AXA^-1) = Φ(A) ϕ(X) Φ(A)^-1, for all X ∈ g, A ∈ G.

(2) ϕ([X,Y]) = [ ϕ(X),  ϕ(Y)], for all  X, Y ∈ g (Lie algebra homomorphism)

(3) ϕ(X) = d/dt Φ(e^tX) |_t=0, for all X ∈ g

(B) Lie group homomorphism

Let G and H be matrix Lie groups with Lie algebras  g and h. Let ϕ :   g --> h be a Lie algebra homomorphism. If G is simply connected , then there exists a unique Lie group homomorphism Φ : G --> H such that  Φ(e^X) = e^ϕ(X) for all X ∈ g. However, if G is not simply connected, this will not be true. We should look for Ğ that has the same Lie algebra as G but such that Ğ is simply connected. 

Baker-Campbell-Hausdorff formula says that if X and Y are sufficiently small, then 

log(e^Xe^Y) = X + Y + 1/2[X,Y] + 1/12[X,[X,Y]] - 1/12[Y,[X,Y]] + ...

Because ϕ is a Lie algebra homomorphism, 

ϕ(log(e^Xe^Y)) = ϕ(X) + ϕ(Y) + 1/2[ϕ(X),ϕ(Y)] + 1/12[ϕ(X),[ϕ(X),ϕ(Y)]] - 1/12[ϕ(Y),[ϕ(X),ϕ(Y)]] + ... = log(e^ϕ(X)e^ϕ(Y))

For group homomorphism, Φ(e^Xe^Y) = exp(ϕ(e^Xe^Y) = exp(ϕ(exp(log(e^Xe^Y) ))). From above, Φ(e^Xe^Y) = exp(log(e^ϕ(X)e^ϕ(Y))) = e^ϕ(X)e^ϕ(Y) = Φ(e^X)Φ(e^Y). Therefore, by Baker-Campbell-Hausdorff formula, if X is small, Φ is a group homomorphism. 

Covering Groups

Let G be a connected Lie group. Then, a universal covering group (or universal cover) of G is a simply-connected Lie group H together with with a Lie group homomorphism Φ : H --> G such that the associated Lie algebra homomorphism ϕ :   h --> g is a Lie algebra homomorphism. The homomorphism Φ is called the covering homomorphism (or projection map). 

For any connected Lie group, a universal cover exists. If G is a connected Lie group and (H₁, Φ₁) and (H₂, Φ₂) are universal covers of G, then there exists a Lie group isomorphism Ψ : H₁ --> H₂ such that Φ₂ o Ψ =  Φ₁. 

The universal cover of a matrix Lie group may not be a matrix Lie group.