Advertisement
Published in:
2012 | OriginalPaper | Chapter
20. The Exponential Map
Activate our intelligent search to find suitable subject content or patents.
Select sections of text to find matching patents with Artificial Intelligence. powered by
Select sections of text to find additional relevant content using AI-assisted search. powered by
Abstract
In this chapter we introduce the exponential map of a Lie group, which is a canonical smooth map from the Lie algebra into the group, mapping lines through the origin in the Lie algebra to one-parameter subgroups. As our first application, we prove the closed subgroup theorem, which says that every topologically closed subgroup of a Lie group is actually an embedded Lie subgroup. Next we prove a higher-dimensional generalization of the fundamental theorem on flows: if G is a simply connected Lie group, then any Lie algebra homomorphism from its Lie algebra into the set of complete vector fields on a smooth manifold M generates a smooth action of G on M. Using this theorem, we prove that there is a one-to-one correspondence between isomorphism classes of finite-dimensional Lie algebras and isomorphism classes of simply connected Lie groups. At the end of the chapter, we show that connected normal subgroups of a Lie group correspond to ideals in its Lie algebra.