# Undecidability of the first order theories of free non-commutative Lie algebras

## TL;DR

This paper proves that the elementary theory of free non-commutative Lie algebras over certain rings is undecidable by interpreting arithmetic within them, answering longstanding questions in their model theory.

## Contribution

It demonstrates that the ring and its action are interpretable in free Lie algebras, leading to undecidability results for their elementary theories.

## Key findings

- The ring R and its action are 0-interpretable in L.
- If R has characteristic zero, Th(L) is undecidable.
- Arithmetic is interpretable in L, showing the independence property.

## Abstract

Let $R$ be a commutative integral unital domain and $L$ a free non-commutative Lie algebra over $R$. In this paper we show that the ring $R$ and its action on $L$ are 0-interpretable in $L$, viewed as a ring with the standard ring language $+, \cdot,0$. Furthermore, if $R$ has characteristic zero then we prove that the elementary theory $Th(L)$ of $L$ in the standard ring language is undecidable. To do so we show that the arithmetic ${\bf N} = \langle{\bf N}, +,\cdot,0 \rangle$ is 0-interpretable in $L$. This implies that the theory of $Th(L)$ has the independence property. These results answer some old questions on model theory of free Lie algebras.