Isomorphisms and elementary equivalence of Chevalley groups over commutative rings

Bunina, E. I.

doi:10.1070/sm9069

Cited by 12 publications

(6 citation statements)

References 17 publications

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“…Continuation of investigations in this field were the papers of Bunina -2010. Similar to Maltsev's results were obtained not only for classical linear groups GL , PGL , SL , PSL , but also for unitary linear groups over fields, skewfields, and rings with involutions (see [6], [7]), for Chevalley groups over fields ( [8]), over local rings (see [9]) and arbitrary commutative rings (see [10]), and also for other different derivative structures.…”

Section: Maltsev-type Theorems For Linear Groupssupporting

confidence: 67%

Elementary equivalence of stable linear groups over fields of characteristic 2

Bunina¹,

Mikhalëv²,

Soloviev³

2023

Preprint

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Section: Maltsev-type Theorems For Linear Groupssupporting

confidence: 67%

Elementary equivalence of stable linear groups over fields of characteristic 2

Bunina¹,

Mikhalëv²,

Soloviev³

2023

Preprint

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“…Then root systems and weight lattices of G1 and G2 coincide, while the rings are elementarily equivalent. In other words Chevalley groups over local rings are elementarily rigid in the class of such groups modulo rigidity of the ground rings [11]. • Suppose Gπ(Φ, K) be simple Chevalley group over the algebraically closed field K.…”

Section: Proposition 210 ([80]mentioning

confidence: 99%

Nonstandard analysis, deformation quantization and some logical aspects of (non)commutative algebraic geometry

Kanel-Belov,

Chilikov,

Ivanov-Pogodaev

et al. 2020

Preprint

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The connections between algebraic geometry and mathematical logic are extremely important. This short survey is related to ideas that contained in the well-known works of B. Plotkin and V. Remeslennikov and their followers. We assume that lots of questions still require further illumination. One of the goals of this paper is to narrow the gap and to draw attention to this topic.We start from a brief review of the paper and motivations. First sections deal with model theory. In Section 2.1 we describe the geometric equivalence, the elementary equivalence, and the isotypicity of algebras. We look at these notions from the positions of universal algebraic geometry and make emphasis on the cases of the first order rigidity. In this setting Plotkin's problem on the structure of automorphisms of (auto)endomorphisms of free objects, and auto-equivalence of categories is pretty natural and important. Section 2.2 is dedicated to particular cases of Plotkin's problem. Section 2.3 is devoted to Plotkin's problem for automorphisms of the group of polynomial symplectomorphisms. This setting has applications to mathematical physics through the use of model theory ( non-standard analysis) in the studying of homomorphisms between groups of symplectomorphisms and automorphisms of the Weyl algebra.The last two sections deal with algorithmic problems for noncommutative and commutative algebraic geometry. Section 3.1 is devoted to the Gröbner basis in non-commutative situation. Despite the existence of an algorithm for checking equalities, the zero divisors and nilpotency problems are algorithmically unsolvable. Section 3.2 is connected with the problem of embedding of algebraic varieties; a sketch of the proof of its algorithmic undecidability over a field of characteristic zero is given. Dedicated to the 70-th anniversary of A.L. Semenov and to the 95-th anniversary of B.I. Plotkin. Contents KANEL-BELOV, CHILIKOV, IVANOV-POGODAEV, MALEV, PLOTKIN, YU, AND ZHANG 3.1. Finite Gröbner basis algebras with unsolvable nilpotency problem and zero divisors problem 22 3.2. On the Algorithmic Undecidability of the Embeddability Problem for Algebraic Varieties over a Field of Characteristic Zero 25 References 27

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“…Then root systems and weight lattices of G 1 and G 2 coincide, while the rings are elementarily equivalent. In other words Chevalley groups over local rings are elementarily rigid in the class of such groups modulo elementary equivalence of ground rings [4].…”

Section: Chevalley Groupsmentioning

confidence: 99%

On first order rigidity for linear groups

Plotkin

2020

Preprint

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Isomorphisms and elementary equivalence of Chevalley groups over commutative rings

Abstract: It is proved that two Chevalley groups with indecomposable root systems of rank over commutative rings (which contain in addition for the types , , , … Show more

Cited by 12 publications

References 17 publications

Elementary equivalence of stable linear groups over fields of characteristic 2

Elementary equivalence of stable linear groups over fields of characteristic 2

Nonstandard analysis, deformation quantization and some logical aspects of (non)commutative algebraic geometry

On first order rigidity for linear groups

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