John Stillwell - Naive Lie Theory.pdf - Index of

9.6 Lifting a Lie algebra homomorphism 197
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Figure 9.4: Sequence deforming the bottom edge to the top.
image of the (i, j)-subsquare. Intuitively speaking, d ◦ d ij moves one side
of the image subsquare to the other, while keeping the image fixed outside
a neighborhood of the image subsquare.
It follows that if d is a deformation of path p to path q and d ij runs
through the sequence of maps that deform the bottom edge of the unit
square to the top, then the sequence of composite maps d ◦ d ij deforms
ptoq,andeachd◦ d ij agrees with d outside a neighborhood of the image
of the (i, j)-subsquare, and hence outside an ε-ball.
In this sense, if a path p can be deformed to a path q, then p can be
deformed to q in a finite sequence of “small” steps.
Exercises
9.5.1 If a < 0 < 1 < b, give a continuous map of (a,b) onto (a,b) that sends 0 to
1. Use this map to define d ij when the (i, j)-subsquare is in the interior of
the unit square.
9.5.2 If 1 < b give a continuous map of [0,b) onto [1,b) that sends 0 to 1, and
use it (and perhaps also the map in Exercise 9.5.1) to define d ij when the
(i, j)-subsquare is one of the boundary squares of the unit square.
9.6 Lifting a Lie algebra homomorphism
Now we are ready to achieve the main goal of this chapter: showing that
if g and h are the Lie algebras of simply connected Lie groups G and H,
respectively, then each Lie algebra homomorphism ϕ : g → h is induced by
a Lie group homomorphism Φ : G → H. This is the converse of the theorem
in Section 9.3, and the two theorems together show that the structure of
simply connected Lie groups is completely captured by their Lie algebras.
The idea of the proof is to “lift” the homomorphism ϕ from g to G in small
pieces, with the help of the exponential function and the Campbell–Baker–
Hausdorff theorem of Section 7.7.