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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.

An important theorem of Cartan asserts that any two maximal tori in a compact Lie group are conjugate. There are different ways of proving this. We will deduce it from the surjectivity of the exponential map, which we will prove by showing that a geodesic between the origin and an arbitrary point of the group has the form \(t\mapsto {\mathrm{e}}^{tX}\) for some X in the Lie algebra.

The exponential map, introduced for closed Lie subgroups of \(\mathrm{GL}(n, \mathbb{C})\) in Chap. 5, can be defined for a general Lie group G as a map Lie(G) → G.

The inventors of Lie groups and Lie algebras (starting with Lie!) regarded Lie groups as groups of symmetries of various topological or geometric objects. Lie algebras were viewed as the “infinitesimal transformations” associated with the symmetries in the Lie group. For example, the group SO(n) of rotations is the group of orientation-preserving isometries of the Euclidean space \( \mathbb{E}^n \) . The Lie algebra so (n,ℝ) consisting of real skew symmetric n × n matrices is the corresponding set of infinitesimal rotations. The geometric link between a Lie group and its Lie algebra is the fact that the Lie algebra can be viewed as the tangent space to the Lie group at the identity. There is a map from the tangent space to the Lie group, called the exponential map. The Lie algebra can be considered as a linearization of the Lie group (near the identity element), and the exponential map provides the “delinearization,” i.e., it takes us back to the Lie algebra. These concepts have a concrete realization in the case of groups of matrices, and for this reason we begin by studying the behavior of the exponential maps on matrices.

This section contains a brief introduction to matrix Lie groups. The focus is on developing tools for the construction of factorizations of a group with respect to some subgroup. In this connection the Lie algebra and the exponential map are the central concepts. It may be noted that all the groups occurring in the examples considered in these notes are matrix Lie groups.