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Preprint Number 1211

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1211. Olga Kharlampovich, Alexei Myasnikov
Undecidability of the first order theories of free non-commutative Lie algebras

Submission date: 25 April 2017


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 { +, ⋅ , 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 N = < N , +, ⋅ , 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.

Mathematics Subject Classification: 03C60

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Full text arXiv 1704.07853: pdf, ps.

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