Undecidability of Equations in Free Lie Algebras

TL;DR

This paper proves that solving finite systems of equations in free Lie algebras of rank three or more over any field is undecidable, by interpreting the ring of integers within these algebras.

Contribution

It establishes the undecidability of equations in free Lie algebras of rank at least three, extending known results to a broader algebraic context.

Findings

Undecidability of equations in free Lie algebras of rank ≥ 3

Interpretation of the ring of integers within these algebras

Results hold over arbitrary fields, including characteristic zero

Abstract

In this paper we prove undecidability of finite systems of equations in free Lie algebras of rank at least three over an arbitrary field. We show that the ring of integers $\mathbb{Z}$ is interpretable by positive existential formulas in such free Lie algebras over a field of characteristic zero.