Lie Groups

Abstract

This chapter discusses problems on Lie groups, Lie algebras and homogeneous spaces. After considering some specific examples of Lie groups and Lie algebras and some questions on them, we consider homomorphisms, Lie subgroups and Lie subalgebras, integration on Lie groups, the exponential map exp and its differential map exp_*, the adjoint representation Ad and its differential map ad, and Lie groups of transformations. We then consider homogeneous spaces, considering some classical examples, as the sphere, the real Stiefel manifold, and the real Grassmannian. Some miscellaneous topics are moreover considered.

We give some problems intended to familiarise the reader with some basic problems of Lie groups and Lie algebra, as the determination of all the 2-dimensional Lie algebras, all (up to isomorphism) 1-dimensional Lie groups, some calculations to see the reason of the name “exponential,” the dimensions of some classical groups, and some properties of the Heisenberg group.