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

Olga Kharlampovich; Alexei Myasnikov

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arXiv · 1704.07853

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

Olga Kharlampovich·Alexei Myasnikov

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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.

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BibTeX RIS

Olga Kharlampovich, Alexei Myasnikov. 2017-04-25. Undecidability of the first order theories of free non-commutative Lie algebras. https://arxiv.org/abs/1704.07853

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