Lie Groups, Lie Algebras, and Representations

Abstract

An important concept in physics is that of symmetry, whether it be rotational symmetry for many physical systems or Lorentz symmetry in relativistic systems. In many cases, the group of symmetries of a system is a continuous group, that is, a group that is parameterized by one or more real parameters. More precisely, the symmetry group is often a Lie group, that is, a smooth manifold endowed with a group structure in such a way that operations of inversion and group multiplication are smooth. The tangent space at the identity in a Lie group has a natural “bracket” operation that makes the tangent space into a Lie algebra. The Lie algebra of a Lie group encodes many of the properties of the Lie group, and yet the Lie algebra is easier to work with because it is a linear space.