5 - Algebraic structure of Lie groups

Summary

This survey of the algebraic structure of Lie groups and Lie algebras (mainly semi simple) is a considerably expanded version of the oral lectures at the symposium. It is limited to what is necessary for representation theory, which is another way of saying that very little has been left out. In spite of its length, it contains few proofs or even indications of proofs, nor have I given chapter and verse for each of the multitude of unproved assertions throughout the text. Instead, I have appended references to each section, from which the diligent reader should have no difficulty in tracking down the proofs.

Lie Groups and Lie Algebras

Vector fields

Let M be a smooth (C^∞) manifold, and for each point x ∈ M let T_x(M) denote the vector space of tangent vectors to M at x. The union of all the T_x(M) is the tangent bundle T(M) of M, Locally, if U if a coordinate neighbourhood in M, the restriction of T(M) to U is just U × Rⁿ, where n is the dimension of M. Each smooth map ϕ: M → N, where N is another smooth manifold, gives rise to a tangent map T(ϕ) : T(M) → T(N), whose restriction T_x(M) to the tangent space T_x(M) is a linear mapping of T_x(M) into T_ϕ(x), (N). In terms of local coordinates in M and N, T_x(ϕ) is given by the x Jacobian matrix.