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Preprint Number 1841

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1841. Hugo Luiz Mariano, João Schwarz
First-order characterization of noncommutative birational equivalence

Submission date: 7 September


Let Σ be a root system with Weyl group W. Let k be an algebraically closed field of zero characteristic, and consider the corresponding semisimple Lie algebra g_{k, Σ}. Then there is a first-order sentence φ_Σ in the language L=(1,0,+,*) of rings sucht that, for any algebraically closed field k of char = 0, the validity of the Gelfand-Kirillov Conjecture for g_{k, Σ} is equivalent to ACF_0 ⊢ φ_Σ. By the same method, we can show that the validity of Noncommutative Noether's Problem for A_n(k)^W, k any algebraically closed field of char = 0 is equivalent to ACF_0 ⊢ φ_W, φ_W a formula in the same language. As consequences, we obtain results on the modular Gelfand-Kirillov Conjecture and we show that, for 𝔽 algebraically closed with characteristic p ≫ 0, A_n(𝔽)^W is a case of positive solution of modular Noncommutative Noether's Problem.

Mathematics Subject Classification: Primary: 03C60, Secundary: 16S85, 16W22, 17B35

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Full text arXiv 2009.03387: pdf, ps.

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