Structure of Classical Groups

Abstract

In this chapter we study the structure of a classical group G and its Lie algebra. We choose a matrix realization of G such that the diagonal subgroup H ⊂ G is a maximal torus; by elementary linear algebra every conjugacy class of semisimple elements intersects H. Using the unipotent elements in G, we show that the groups GL(n,$\mathbb{C}$), SL(n,$\mathbb{C}$), SO(n,$\mathbb{C}$), and Sp(n,$\mathbb{C}$) are connected (as Lie groups and as algebraic groups). We examine the group SL(2,C), find its irreducible representations, and show that every regular representation decomposes as the direct sum of irreducible representations. This group and its Lie algebra play a basic role in the structure of the other classical groups and Lie algebras. We decompose the Lie algebra of a classical group under the adjoint action of a maximal torus and find the invariant subspaces (called root spaces) and the corresponding characters (called roots). The commutation relations of the root spaces are encoded by the set of roots; we use this information to prove that the classical (trace-zero) Lie algebras are simple (or semisimple). In the final section of the chapter we develop some general Lie algebra methods (solvable Lie algebras, Killing form) and show that every semisimple Lie algebra has a root-space decomposition with the same properties as those of the classical Lie algebras.