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

Abstract

Let $D$ be a division algebra such that $D\t D^o$ is a Noetherian algebra, then any division subalgebra of $D$ is a {\em finitely generated} division algebra. Let $\D $ be a finite set of commuting derivations or automorphisms of the division algebra $D$, then the group $\Ev (\D)$ of common eigenvalues (i.e. {\em weights}) is a {\em finitely generated abelian} group. Typical examples of $D$ are the quotient division algebra ${\rm Frac} (\CD (X))$ of the ring of differential operators $\CD (X)$ on a smooth irreducible affine variety $X$ over a field $K$ of characteristic zero, and the quotient division algebra ${\rm Frac} (U (\Gg))$ of the universal enveloping algebra $U(\Gg)$ of a finite dimensional Lie algebra $\Gg $. It is proved that the algebra of differential operators $\CD (X)$ is isomorphic to its opposite algebra $\CD (X)^o$.