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

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 $\mathbf{N}=⟨\mathbf{N},+,\cdot ,0⟩$ 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.

Misprints corrected