`New' examples of skew fields not finitely generated as algebras

TL;DR

This paper investigates the conditions under which certain division algebras, especially those arising from skew polynomial rings and quantum groups, are affine or nonaffine over their centers, extending classical results.

Contribution

It provides new examples of division algebras that are not finitely generated as algebras over their centers, especially in the context of skew polynomial rings and quantum groups.

Findings

Many transcendental division algebras are nonaffine over their centers.

Division algebras of fractions of Weyl algebras are affine only when finite dimensional.

Identifies conditions for affineness in division algebras from quantum spaces.

Abstract

An associative division algebra D is said to be _affine_ over a central subfield k if D is finitely generated as a k-algebra. In 1956 Amitsur famously proved that, when k is uncountable, D cannot be k-affine unless D is algebraic over k. In this paper we consider affineness -- and nonaffineness -- for certain naturally occurring classes of division algebras over arbitrary fields. The primary applications are to division algebras of fractions of suitably conditioned iterated skew polynomial rings over k, including many examples naturally arising in Lie theoretic and quantum group settings. Many transcendental division algebras are thus verified to be nonaffine over k. Division algebras of fractions of Weyl algebras and quantum affine spaces are determined to be affine over their centers exactly when they are finite dimensional over their centers.