Introduction to Lie Groups

Abstract

Roughly speaking, a Lie group is a “group” which is also a “manifold”. Of course, to make sense of this definition, we must explain these two basic concepts and how they can be related. Groups arise as an algebraic abstraction of the notion of symmetry; an important example is the group of rotations in the plane or three-dimensional space. Manifolds, which form the fundamental objects in the field of differential geometry, generalize the familiar concepts of curves and surfaces in three-dimensional space. In general, a manifold is a space that locally looks like Euclidean space, but whose global character might be quite different. The conjunction of these two seemingly disparate mathematical ideas combines, and significantly extends, both the algebraic methods of group theory and the multi-variable calculus used in analytic geometry. This resulting theory, particularly the powerful infinitesimal techniques, can then be applied to a wide range of physical and mathematical problems.