From hydrogen atom to generalized Dynkin diagrams

We identify the “dynamical algebras" HD of the D-dimensional hydrogen atom as positive subalgebras of twisted and untwisted affine Kac-Moody algebras: For odd D≥5 we obtain H 2l+1≃Dl+1(2)+. But for even D≥6, H 2l is a parabolic subalgebra of Bl(1). H 4 is a parabolic subalgebra of C2(1), H 3≃D2(2)+≃A1(1)+, while H 2 is isomorphic to the Borel subalgebra of A1(1). Along the way we prove a theorem on the untwisting of positive subalgebras of twisted affine algebras, and introduce generalized Dynkin diagrams which enable us to represent graphically automorphisms and parabolic subalgebras of finite and affine algebras.