John Stillwell - Naive Lie Theory.pdf - Index of

9
Simply connected Lie groups
PREVIEW
Throughout our exposition of Lie algebras we have claimed that the structure
of the Lie algebra g of a Lie group G captures most, if not all, of the
structure of G. Now it is time to explain what, if anything, is lost when we
pass from G to g. The short answer is that topological information is lost,
because the tangent space g cannot reveal how G may “wrap around” far
from the identity element.
The loss of information is already apparent in the case of R,O(2), and
SO(2), all of which have the line as tangent space. A more interesting case
is that of O(3),SO(3), andSU(2), all of which have the Lie algebra so(3).
These three groups are not isomorphic, and the differences between them
are best expressed in topological language, because the differences persist
even if we distort O(3), SO(3), andSU(2) by continuous 1-to-1 maps.
First, O(3) differs topologically from SO(3) and SU(2) because it is
not path-connected; there are two points in O(3) not connected by a path
in O(3). Second, SU(2) differs topologically from SO(3) in being simply
connected; that is, any closed path in SU(2) can be shrunk to a point.
We elaborate on these properties of O(3), SO(3), andSU(2) in Sections
9.1 and 9.2. Then we turn to the relationship between homomorphisms
of Lie groups and homomorphisms of Lie algebras: a Lie group homomorphism
Φ : G → H “induces” a Lie algebra homomorphism ϕ : g → h
and if G and H are simply connected then ϕ uniquely determines Φ. This
leads to a definitive result on the extent to which a Lie algebra g “determines”
its Lie group G: all simply connected groups with the same Lie
algebra are isomorphic.
186 J. Stillwell, Naive Lie Theory, DOI: 10.1007/978-0-387-78214-0 9,
c○ Springer Science+Business Media, LLC 2008