Division algebras of Gelfand-Kirillov dimension two

Abstract

Let $A$ be a finitely generated $K$-algebra that is a domain of GK dimension less than 3, and let $Q(A)$ denote the quotient division algebra of $A$. We show that if $D$ is a division subalgebra of $Q(A)$ of GK dimension at least 2 then $Q(A)$ is finite dimensional as a left $D$-vector space. We use this to show that if $A$ is a finitely generated domain of GK dimension less than 3 over an algebraically closed field $K$ then any division subalgebra $D$ of $Q(A)$ is either a finitely generated field extension of $K$ of transcendence degree at most one, or $Q(A)$ is finite dimensional as a left $D$-vector space.