Finite generation of division subalgebras and of the group of eigenvalues for commuting derivations or automorphisms of division algebras

Let D be a division algebra such that D ⊗ D is a Noetherian algebra, then any division subalgebra of D is a finitely generated division algebra. Let ∆ be a finite set of commuting derivations or automorphisms of the division algebra D, then the group Ev(∆) 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. It is proved that the algebra of differential operators D(X) is isomorphic to its opposite algebra D(X). Mathematics subject classification 2000: 16S15, 16W25, 16S32, 16P40, 16K40.