Finite Generation of the Group of Eigenvalues for Sets of Derivations or Automorphisms of Division Algebras

Abstract

Let D be a division algebra such that D⊗ D o is a Noetherian algebra, and Δ be a set of derivations or automorphisms of the division algebra D, then the group Ev(Δ) of common eigenvalues (i.e., weights) is a finitely generated abelian group. Typical examples of D are the quotient division algebra Frac (𝒟 (X)) of the ring of differential operators 𝒟 (X) on a smooth irreducible affine variety X over a field K of characteristic zero, and the quotient division algebra Frac (U (𝔤)) of the universal enveloping algebra U(𝔤) of a finite dimensional Lie algebra 𝔤.