Elementary equivalence of Kac-Moody groups

TL;DR

This paper investigates the model-theoretic properties of Kac-Moody groups, establishing conditions under which elementary equivalence implies structural similarities such as identical Cartan matrices and ground fields.

Contribution

It provides new results linking elementary equivalence of Kac-Moody groups to their underlying algebraic and combinatorial data, including Cartan matrices and fields.

Findings

Elementary equivalence of affine Kac-Moody groups implies identical Cartan matrices.

Elementary equivalence over finite fields implies identical fields and twin root data.

Results extend to Kac-Moody groups over infinite subfields of algebraic closures of finite fields.

Abstract

The paper is devoted to model-theoretic properties of Kac-Moody groups with the focus on elementary equivalence of Kac-Moody groups. We show that elementary equivalence of (untwisted) affine Kac-Moody groups implies coincidence of their generalized Cartan matrices and the elementary equivalence of their ground fields. We also show that elementary equivalence of arbitrary Kac-Moody groups over finite fields implies coincidence of these fields and an isomorphism of their twin root data. The similar result is established for Kac-Moody groups defined over infinite subfields of the algebraic closures of finite fields.