Lie Groups and Lie Algebras

A Lie group is, roughly speaking, an analytic manifold with a group structure such that the group operations are analytic. Lie groups arise in a natural way as transformation groups of geometric objects. For example, the group of all affine transformations of a connected manifold with an affine connection and the group of all isometries of a pseudo-Riemannian manifold are known to be Lie groups in the compact open topology. However, the group of all diffeomorphisms of a manifold is too big to form a Lie group in any reasonable topology. The tangent space g at the identity element of a Lie group G has a rule of composition (X,Y) -[X,Y] derived from the bracket operation on the left invariant vector fields on G. The vector space g with this rule of composition is called the Lie algebra of G. The structures of g and G are related by the exponential mapping exp: g G which sends straight lines through the origin in g onto one-paramater subgroups of G. Several properties of this mapping are developed already in §1 because they can be derived as special cases of properties of the Exponential mapping for a suitable affine connection on G. Although the structure of g is determined by an arbitrary neighborhood of the identity element of G, the exponential mapping sets up a far-reaching relationship between g and the group G in the large. We shall for example see in Chapter VII that the center of a compact simply connected Lie group G is explicitly determined by the Lie algebra g. In §2 the correspondence (induced by exp) between subalgebras and subgroups is developed. This correspondence is of basic importance in the theory in spite of its weakness that the subalgebra does not in general decide whether the corresponding subgroup will be closed or not, an important distinction when coset spaces are considered. In §4 we investigate the relationship between homogeneous spaces and coset spaces. It is shown that if a manifold M has a separable transitive Lie transformation group G acting on it, then M can be identified with a coset space GIH (H closed) and therefore falls inside the realm of Lie group theory. Thus, one can, for example, conclude that if H is compact, then M has a G-invariant Riemannian structure. Let G be a connected Lie group with Lie algebra g. If ao G, the inner automorphism g ogol induces an automorphism Ad (a) of g and the mapping a Ad (a) is an analytic homomorphism of G onto an analytic subgroup Ad (G) of GL(g), the adjoint group. The group Ad (G) can be defined by g alone and since its Lie algebra is isomorphic to g/3 ( = center of g), one can, for example, conclude that a semisimple Lie algebra over R is isomorphic to the Lie algebra of a Lie group. This fact holds for arbitrary Lie algebras over R but will not be needed in this book in that generality. Section 6 deals with some preliminary results about semisimple Lie groups. The main result is Weyl's theorem stating that the universal covering group of a compact semisimple Lie group is compact. In §7 we discuss invariant forms on G and their determination from the structure of g.