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  • Print publication year: 1987
  • Online publication date: August 2012

12 - Lie groups

Summary

A group whose elements are labelled by one or more continuously variable parameters may be considered also to be a manifold; one has merely to take the parameters as coordinates. This is the basic idea of the theory of Lie groups. The groups in question might well have been called differentiable groups, but the conventional association with the name of Sophus Lie, who revolutionised the theory of differentiable groups in the last decades of the nineteenth century, is too deeply ingrained in the literature to admit any change.

Many examples of Lie groups have already arisen in this book. The affine group introduced in Chapter 1 is a Lie group. So also are the rotation, Euclidean, Lorentz and Poincaré groups of Chapter 8. The one-parameter groups of transformations introduced in Chapter 3 are (1-dimensional) Lie groups.

The discussion of these groups in this chapter differs in emphasis from that of the preceding chapters. The groups just mentioned arose as groups of transformations of other manifolds. We have hinted already that one can abstract the group structure from the idea of a transformation group and consider the group in its own right without regard to the manifold on which it acts. One can go further than this, and define a Lie group abstractly in the first place, as a manifold endowed with maps defining group multiplication and formation of inverses. This is how the definition is usually presented nowadays.