Lie Groups
In this chapter we introduce Lie groups, which are smooth manifolds that are also groups in which multiplication and inversion are smooth maps. Besides providing many examples of interesting manifolds themselves, they are essential tools in the study of m
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Lie Groups
In this chapter we introduce Lie groups, which are smooth manifolds that are also groups in which multiplication and inversion are smooth maps. Besides providing many examples of interesting manifolds themselves, they are essential tools in the study of more general manifolds, primarily because of the role they play as groups of symmetries of other manifolds. Our aim in this chapter is to introduce Lie groups and some of the tools for working with them, and to describe an abundant supply of examples. In subsequent chapters (especially Chapters 8, 20, and 21), we will develop many more properties and applications of Lie groups. We begin with the definition of Lie groups and some of the basic structures associated with them, and then present a number of examples. Next we study Lie group homomorphisms, which are group homomorphisms that are also smooth maps. Then we introduce Lie subgroups (subgroups that are also smooth submanifolds), which lead to a number of new examples of Lie groups. After explaining these basic ideas, we introduce actions of Lie groups on manifolds, which are the primary raison d’être of Lie groups. At the end of the chapter, we briefly touch on group representations. The study of Lie groups was initiated in the late nineteenth century by the Norwegian mathematician Sophus Lie. Inspired by the way the French algebraist Évariste Galois had invented group theory and used it to analyze polynomial equations, Lie was interested in using symmetries, expressed in the form of group actions, to simplify problems in partial differential equations and geometry. However, Lie could not have conceived of the global objects that we now call Lie groups, for the simple reason that global topological notions such as manifolds (or even topological spaces!) had not yet been formulated. What Lie studied was essentially a localcoordinate version of Lie groups, now called local Lie groups. Despite the limitations imposed by the era in which he lived, he was able to lay much of the groundwork for our current understanding of Lie groups. We will describe his principal results in Chapter 20 (see Theorem 20.16). J.M. Lee, Introduction to Smooth Manifolds, Graduate Texts in Mathematics 218, DOI 10.1007/978-1-4419-9982-5_7, © Springer Science+Business Media New York 2013
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Basic Definitions
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Basic Definitions A Lie group is a smooth manifold G (without boundary) that is also a group in the algebraic sense, with the property that the multiplication map mW G G ! G and inversion map i W G ! G, given by m.g; h/ D gh;
i.g/ D g 1 ;
are both smooth. A Lie group is, in particular, a topological group (a topological space with a group structure such that the multiplication and inversion maps are continuous). The group operation in an arbitrary Lie group is denoted by juxtaposition, except in certain abelian groups such as Rn in which the operation is usually written additively. It is traditional to denote the identity element of an arbitrary Lie group by the symbol e (for German Einselement, “unit element”
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