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

Research paper by **V. V. Bavula**

Indexed on: **06 May '05**Published on: **06 May '05**Published in: **Mathematics - Rings and Algebras**

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#### 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 $\CD (X)^o$.