Basics of Classical Lie Groups: The Exponential Map, Lie Groups, and Lie Algebras

The inventors of Lie groups and Lie algebras (starting with Lie!) regarded Lie groups as groups of symmetries of various topological or geometric objects. Lie algebras were viewed as the “infinitesimal transformations” associated with the symmetries in the Lie group. For example, the group SO(n) of rotations is the group of orientation-preserving isometries of the Euclidean space \( \mathbb{E}^n \) . The Lie algebra so (n,ℝ) consisting of real skew symmetric n × n matrices is the corresponding set of infinitesimal rotations. The geometric link between a Lie group and its Lie algebra is the fact that the Lie algebra can be viewed as the tangent space to the Lie group at the identity. There is a map from the tangent space to the Lie group, called the exponential map. The Lie algebra can be considered as a linearization of the Lie group (near the identity element), and the exponential map provides the “delinearization,” i.e., it takes us back to the Lie algebra. These concepts have a concrete realization in the case of groups of matrices, and for this reason we begin by studying the behavior of the exponential maps on matrices.