Lie Groups
In this chapter, we give an introduction to the theory of (real) Lie groups. We start with the basic notions and a number of examples, including the classical groups. Thereafter, we discuss left-invariant vector fields, define the Lie algebra of a Lie gro
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Lie Groups
In this chapter, we give an introduction to the theory of Lie groups. In Sect. 5.1, we discuss the basic notions and provide the reader with a number of examples. In particular, we take up the classical groups which have already been introduced in Sect. 1.2. Next, in Sect. 5.2, we come to the notion of Lie algebra of a Lie group. We consider a number of examples, again with some emphasis on the Lie algebras of classical Lie groups. Section 5.3 is devoted to an important tool, the exponential mapping. This mapping constitutes a link between the Lie group and its Lie algebra which turns out to be useful both for the study of the local structure of Lie groups and for the study of their representations. Next, in Sect. 5.4, we discuss a number of important representations—the adjoint and the coadjoint representations of the Lie group and the corresponding derived representations of its Lie algebra. Using the adjoint representation of the Lie algebra, one can construct a natural symmetric bilinear form on the Lie algebra, the Killing form, which is invariant under the adjoint representation of the group. Next, in Sect. 5.5, we discuss the concept of left-invariant forms which in particular yields a unique (up to multiplication by a number) left-invariant volume form on every Lie group. The latter gives rise to the so-called Haar measure. We discuss the relation with Ad-invariant scalar products on the Lie algebra and conclude, in particular, that every compact Lie group admits a bi-invariant Riemannian metric. The final two sections are devoted to the theory of Lie subgroups and to homogeneous spaces. Concerning the latter, we discuss three important examples in detail: Stiefel manifolds, Graßmann manifolds and flag manifolds.
5.1 Basic Notions and Examples The notion of Lie group arises naturally by combining the algebraic structure of a group with the differentiable structure of a smooth manifold and requiring that the two structures are compatible. For a group G and elements a, b ∈ G, we denote the product by ab, the unit element by 1 (or sometimes also be e) and the inverse of a by a −1 . The assignment G. Rudolph, M. Schmidt, Differential Geometry and Mathematical Physics, Theoretical and Mathematical Physics, DOI 10.1007/978-94-007-5345-7_5, © Springer Science+Business Media Dordrecht 2013
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5 Lie Groups
a → a −1 defines a bijective mapping inv : G → G, called the inversion mapping of G. Definition 5.1.1 (Lie group) A Lie group is a set which carries the algebraic structure of a group and the differentiable structure of a smooth manifold such that the mapping G × G → G,
(a, b) → ab−1
(5.1.1)
is smooth. A homomorphism of Lie groups is a mapping which is both a group homomorphism and smooth. Remark 5.1.2 1. The inversion mapping g → g −1 is smooth, because it is the restriction of the mapping (5.1.1) to the submanifold {1} × G. Moreover, the multiplication mapping is smooth, because it can be written as the composition of (5.1.1) with the inversion mapping. Using the Inverse Mapping Theor
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