Gelfand-Kirillov conjecture as a first-order formula

Abstract

Let $\Sigma$ be a (reduced) root system. Let $\mathsf{k}$ be an algebraically closed field of zero characteristic, and consider the corresponding semisimple Lie algebra $\mathfrak{g}_{\mathsf{k}, \Sigma}$. Then there is a first-order sentence $\phi_\Sigma$ in the language $\mathcal{L}=(1,0,+,*,-)$ of rings such that, for any algebraically closed field $\mathsf{k}$ of characteristic 0, the validity of the Gelfand-Kirillov Conjecture for $\mathfrak{g}_{\mathsf{k}, \Sigma}$ is equivalent to $ACF_0 \vdash \phi_\Sigma$.