John Stillwell - Naive Lie Theory.pdf - Index of

9.7 Discussion 201
Thus Φ induces the Lie algebra homomorphism ϕ.
Putting these four stages together, we finally have the result:
Homomorphisms of simply connected groups. If g and h are the Lie
algebras of the simply connected Lie groups G and H, respectively, and if
ϕ : g → h is a homomorphism, then there is a homomorphism Φ : G → H
that induces ϕ.
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Corollary. If G and H are simply connected Lie groups with isomorphic
Lie algebras g and h, respectively, then G is isomorphic to H.
Proof. Suppose that ϕ : g → h is a Lie algebra isomorphism, and let the
homomorphism that induces ϕ be Φ : G → H. Also, let Ψ : H → G be the
homomorphism that induces ϕ −1 . It suffices to show that Ψ = Φ −1 ,since
this implies that Φ is a Lie group isomorphism.
Well, it follows from the definition of the “lifted” homomorphisms that
Ψ ◦ Φ : G → G is the unique homomorphism that induces the identity map
ϕ −1 ◦ ϕ : g → g, hence Ψ ◦ Φ is the identity map on G. In other words,
Ψ = Φ −1 .
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9.7 Discussion
The final results of this chapter, and many of the underlying ideas, are due
to Schreier [1925] and Schreier [1927]. In the 1920s, understanding of
the connections between group theory and topology grew rapidly, mainly
under the influence of topologists, who were interested in discrete groups
and covering spaces. Schreier was the first to see clearly that topology is
important in Lie theory and that it separates Lie algebras from Lie groups.
Lie algebras are topologically trivial but Lie groups are generally not, and
Schreier introduced the concept of covering space to distinguish between
Lie groups with the same Lie algebra. He pointed out that every Lie group
G has a universal covering ˜G → G, the unique continuous local isomorphism
of a simply connected group onto G. Examples are the homomorphisms
R → S 1 and SU(2) → SO(3). In general, the universal covering is
constructed by “lifting,” much as we did in the previous section.
The universal covering construction is inverse to the construction of the
quotient by a discrete group because the kernel of ˜G → G is a discrete subgroup
of ˜G, known to topologists as the fundamental group of G, π 1 (G).
Thus G is recovered from ˜G as the quotient ˜G/π 1 (G) =G. Another important
result discovered by Schreier [1925] is that π 1 (G) is abelian for a