The Component Group of the Automorphism Groupof a Simple Lie Algebra and the Splittingof the Corresponding Short Exact Sequence

摘要

Let g be a simple Lie algebra of finite dimension over K ∈ {R, C} and Aut(g) the finite-dimensional Lie group of its automorphisms. We will calculate the component group π_0(Aut(g)) = Aut(g)/ Aut(g)_0, the number of its conjugacy classes and will show that the corresponding short exact sequence 1 → Aut(g)_0 → Aut(g) → π_0(Aut(g))→ 1 is split or, equivalently, there is an isomorphism Aut(g)≈ Aut(g)_0 x π_0(Aut(g)). Indeed, since Aut(g)_0 is open in Aut(g), the quotient group π_0(Aut(g)) is discrete. Hence a section π_0(Aut(g))→ Aut(g) is automatically continuous giving rise to an isomorphism of Lie groups Aut(g)≈ Aut(g)_0 x π_0(Aut(g)).