A two-sorted theory of nilpotent Lie algebras

TL;DR

This paper develops a model-theoretic framework for c-nilpotent Lie algebras over fields, establishing properties like NSOP$_4$ and c-NIP depending on the field's characteristics.

Contribution

It introduces a new model companion for the two-sorted theory of c-nilpotent Lie algebras and analyzes its model-theoretic properties without relying on stationary independence.

Findings

Model companion exists for the theory over certain fields.

If the field is NSOP$_1$, then the model companion is NSOP$_4$.

If the field is algebraically closed, the model companion is c-NIP.

Abstract

We prove the existence of a model companion of the two-sorted theory of $c$-nilpotent Lie algebras over a field satisfying a given theory of fields. We describe a language in which it admits relative quantifier elimination up to the field sort. Using a new criterion which does not rely on a stationary independence relation, we prove that if the field is NSOP$_1$, then the model companion is NSOP$_4$. We also prove that if the field is algebraically closed, then the model companion is $c$-NIP.