Lie groups, named after the Norwegian mathematician Sophus Lie (1842-1899), stand at the intersection of several different branches of mathematics: algebra, analysis, geometry and topology. They are defined as groups that have the additional geometric structure of a smooth manifold. To tie the algebraic structure together with the geometric structure, the operations of multiplication and inversion are required to be differentiable.

At first glance, Lie groups might seem like the mathematical analogue of a centaur--an unnatural combination of two different creatures. But unlike centaurs, Lie groups really do exist. Some examples include:

• The Euclidean group, consisting of all rigid motions of the plane (including rotations, translations and reflections).
• The Lorentz group, consisting of all Lorentz transformations of special relativity that map one inertial reference frame to another.
• The general linear group GL(n), consisting of all n-by-n invertible matrices. The group operation is matrix...