From hydrogen atom to generalized Dynkin diagrams

We identify the "dynamical algebras" ℍ_D 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 ≃ D⁽²⁾⁺_l+1. But for even D ≥ 6, ℍ_2l is a parabolic subalgebra of B⁽¹⁾_l. ℍ₄ is a parabolic subalgebra of C⁽¹⁾₂, ℍ₃ ≃ D⁽²⁾⁺₂ ≃ A⁽¹⁾⁺₁, while ℍ₂ is isomorphic to the Borel subalgebra of A⁽¹⁾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.