Gelfand-Kirillov conjecture as a first-order formula

TL;DR

This paper encodes the Gelfand-Kirillov conjecture for semisimple Lie algebras associated with root systems as a first-order logical statement, linking algebraic properties to model theory.

Contribution

It formulates the Gelfand-Kirillov conjecture as a first-order formula in ring language, connecting algebraic conjectures with logical expressibility.

Findings

The conjecture's validity is equivalent to a first-order sentence in algebraically closed fields.

The approach bridges Lie algebra theory and model theory.

Provides a logical characterization of the Gelfand-Kirillov conjecture.

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 $ACF_0 \vdash \phi_\Sigma$.