The Exponential Map
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 cl
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The Exponential Map
In this chapter we apply the tools of flows, Lie derivatives, and foliations to delve deeper into the relationships between Lie groups and Lie algebras. In the first section we define one-parameter subgroups of a Lie group G, which are just Lie group homomorphisms from R to G, and show that there is a oneto-one correspondence between elements of Lie.G/ and one-parameter subgroups of G. Next we introduce the focal point of our study, which is a canonical smooth map from the Lie algebra into the group called the exponential map. It maps lines through the origin in Lie.G/ to one-parameter subgroups of G. As our first major application of the exponential map, 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. Instead of a single smooth vector field generating an action of R, we consider a finite-dimensional family of vector fields and ask when they generate an action of some Lie group. The main theorem is that if G is a simply connected Lie group, then any Lie algebra homomorphism from Lie.G/ into the set of complete vector fields on M generates a smooth action of G on M . Finally, in the last two sections, we bring together all of these results to deepen our understanding of the correspondence between Lie groups and Lie algebras. First, 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; and then we show that for any Lie group G, connected normal subgroups of G correspond to ideals in the Lie algebra of G, which are subspaces that are stable under bracketing with arbitrary elements of the algebra. This is an excellent illustration of the fundamental philosophy of Lie theory: as much as possible, we use the Lie group/Lie algebra correspondence to translate grouptheoretic questions about a Lie group into linear-algebraic questions about its Lie algebra. J.M. Lee, Introduction to Smooth Manifolds, Graduate Texts in Mathematics 218, 515 DOI 10.1007/978-1-4419-9982-5_20, © Springer Science+Business Media New York 2013
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The Exponential Map
One-Parameter Subgroups and the Exponential Map Suppose G is a Lie group. Since left-invariant vector fields are naturally defined in terms of the group structure of G, one might reasonably expect to find some relationship between the group law for the flow of a left-invariant vector field and group multiplication in G. We begin by exploring this relationship.
One-Parameter Subgroups A one-parameter subgroup of G is defined to be a Lie group homomorphism W R ! G, with R considered as a Lie group under addition. By this definition, a one-parameter subgroup is not a Lie subgroup of G, but rather a homomorphism into G. (However, the image of a one-parameter subgroup is a Lie subgroup when endowed with a suitable smooth manifold structure; see Problem 20-1.) Theorem
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