John Stillwell - Naive Lie Theory.pdf - Index of

188 9 Simply connected Lie groups
Thus we have three Lie groups with the same Lie algebra: O(2),SO(2),
and R. These groups can be distinguished algebraically in various ways
(exercises), but the most obvious differences between them are topological:
• O(2) is not path-connected.
• SO(2) is path-connected but not simply connected, that is, there is a
closed path in SO(2) that cannot be continuously shrunk to a point.
• R is path-connected and simply connected.
Another difference is that both O(2) and SO(2) are compact, that is, closed
and bounded, and R is not.
As this chapter unfolds, we will see that the properties of compactness,
path-connectedness, and simple connectedness are crucial for distinguishing
between Lie groups with the same Lie algebra. These properties are
“squeezed out” of the Lie group G when we form its Lie algebra g, and
we need to put them back in order to “reconstitute” G from g. In particular,
we will see in Section 9.6 that G can be reconstituted uniquely from
g if we know that G is simply connected. But before looking at simple
connectedness more closely, we study another example.
Exercises
9.1.1 Find algebraic properties showing that the groups O(2), SO(2), andR are
not isomorphic.
From the circle group S 1 = SO(2) and the line group R we can construct three
two-dimensional groups as Cartesian products: S 1 × S 1 , S 1 × R,andR × R.
9.1.2 Explain why it is appropriate to call these groups the torus, cylinder, and
plane, respectively.
9.1.3 Show that the three groups have the same Lie algebra. Describe its underlying
vector space and Lie bracket operation.
9.1.4 Distinguish the three groups algebraically and topologically.
9.2 Three groups with the cross-product Lie algebra
At various points in this book we have met the groups O(3), SO(3), and
SU(2), and observed that they all have the same Lie algebra: R 3 with the
cross product operation. Their Lie algebra may also be viewed as the space