The Gelfand–Kirillov Conjecture as a First-Order Formula

We show that for a given (reduced) root system Σ and for any algebraically closed field k with zero characteristic, the validity of the Gelfand–Kirillov conjecture for the finite-dimensional Lie algebra \({\mathfrak{g}}_{\mathrm{k},\Sigma },\) ${\mathfrak{g}}_{\mathrm{k},\mathrm{\Sigma }},$ the sole semisimple Lie algebra over k with root system Σ, is equivalent to the provability of a certain first-order sentence in the language of rings \(\mathcal{L}\) $\mathcal{L}$ (0, 1, +, ∗, −) in the theory ACF₀ of algebraically closed fields of zero characteristic.