Semisimple Compact Groups

Abstract

A Lie algebra is semisimple if it has no Abelian ideals. A compact Lie group is semisimple if its Lie algebra is semisimple. For example, SU(n) and O(n) are semisimple, but U(n) is not, since the scalar matrices in u(n) form an Abelian ideal. More generally, we define a Lie group to be semisimple if its Lie algebra is semisimple and it has a faithful finite-dimensional complex representation. (This criterion excludes groups such as the universal cover of SL(2, ℝ), all of whose finite-dimensional complex representations factor through SL(2, ℝ) and are hence not faithful; see Exercise 13.1.)