The Gelfand–Kirillov Conjecture as a First-Order Formula

IF 0.6 3区数学 Q4 LOGIC

Algebra and Logic Pub Date : 2025-08-26 DOI:10.1007/s10469-025-09792-3

H. L. Mariano, J. Schwarz

引用次数: 0

Abstract

We show that for a given (reduced) root system Σ and for any algebraically closed field k with zero characteristic, the validity of the Gelfand–Kirillov conjecture for the finite-dimensional Lie algebra \({\mathfrak{g}}_{\mathrm{k},\Sigma },\) the sole semisimple Lie algebra over k with root system Σ, is equivalent to the provability of a certain first-order sentence in the language of rings \(\mathcal{L}\)(0, 1, +, ∗, −) in the theory ACF₀ of algebraically closed fields of zero characteristic.

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一阶公式的Gelfand-Kirillov猜想

我们证明了对于给定的（约化）根系统Σ和对于任何具有零特征的代数闭域k，有限维李代数\({\mathfrak{g}}_{\mathrm{k},\Sigma },\) （k上具有根系统Σ的唯一半简单李代数）的Gelfand-Kirillov猜想的有效性等价于零特征代数闭域理论ACF0中环语言\(\mathcal{L}\)（0,1, +,∗,−）中一阶句子的可证明性。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Algebra and Logic 数学-数学

CiteScore

1.10

自引率

20.00%

发文量

审稿时长

>12 weeks

期刊介绍： This bimonthly journal publishes results of the latest research in the areas of modern general algebra and of logic considered primarily from an algebraic viewpoint. The algebraic papers, constituting the major part of the contents, are concerned with studies in such fields as ordered, almost torsion-free, nilpotent, and metabelian groups; isomorphism rings; Lie algebras; Frattini subgroups; and clusters of algebras. In the area of logic, the periodical covers such topics as hierarchical sets, logical automata, and recursive functions. Algebra and Logic is a translation of ALGEBRA I LOGIKA, a publication of the Siberian Fund for Algebra and Logic and the Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences. All articles are peer-reviewed.