2023/11/03 by E. I. Bunina, Bunina, Elena, Pavel Gvozdevsky +1 · 1 citation
Computer Science · Mathematics · #03C60 (Secondary) #20G35 (Primary) 20A15 #FOS: Mathematics #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Logic (math.LO) #Rings, Modules, and Algebras #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2311.01954
openalex publication_date 2023/11/03 · openalex created_date 2023/11/07 · openalex updated_date 2026/07/28
In this paper we consider Chevalley groups over commutative rings with~1, constructed by irreducible root systems of rank >1. We always suppose that for the systems A2, B_ℓ, C_ℓ, F4, G2 our rings contain 1/2 and for the system G2 also 1/3. Under these assumptions we prove that the central quotients of Chevalley groups are regularly bi-interpretable with the corresponding rings, the class of all central quotients of Chevalley groups of a given type is elementarily definable and even finitely axiomatizable (see Definition~2.2). The same holds for adjoint Chevalley groups and for bondedly generated Chevalley groups. We also give an example of Chevalley group with infinite center, which is not bi-interpretable with the corresponding ring and is elementarily equivalent to a group that is not a Chevalley group itself.