Gelfand-Kirillov conjecture as a first-order formula

Abstract

Let $\hat{I}\pounds$ be a (reduced) root system. Let $\mathsf{k}$ be an algebraically closed field of zero characteristic, and consider the corresponding semisimple Lie algebra ${\mathfrak{g}}_{\mathsf{k},\hat{I}\pounds }$. Then there is a first-order sentence $\ddot{I}{•}_{\hat{I}}\pounds$ in the language $\mathcal{L}=(1,0,+,\ast ,-)$ of rings such that, for any algebraically closed field $\mathsf{k}$ of characteristic 0, the validity of the Gelfand-Kirillov Conjecture for ${\mathfrak{g}}_{\mathsf{k},\hat{I}\pounds }$ is equivalent to $AC{F}_0⊢\ddot{I}{•}_{\hat{I}}\pounds$.

The same paper as before, with as an extra section an appendix with some new results that are immediate consequences of v1 of this preprint. V1 of this preprint is still a work in progress