☆ 4.5 Article

Defining R and G.R/

JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY (2023)

Journal

JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY

Volume 25, Issue 8, Pages 3325-3358

Publisher

EUROPEAN MATHEMATICAL SOC-EMS

DOI: 10.4171/JEMS/1255

Keywords

Chevalley groups; bi-interpretation

Categories

Mathematics, Applied Mathematics

Ask authors/readers for more resources

Protocol

Community support

Reagent

Community support

Automated Summary New
Abstract

We prove that each root subgroup of Chevalley groups G(R) of rank at least 2 over an integral domain R is (essentially) the double centralizer of a corresponding root element. This result leads to a new approach to bi-interpretability for algebraic groups over a wide range of rings and fields. It also shows that if R is (finitely) axiomatizable in the corresponding class of rings, then the group G(R) is (finitely) axiomatizable in the appropriate class of groups.

We show that for Chevalley groups G(R) of rank at least 2 over an integral domain R each root subgroup is (essentially) the double centralizer of a corresponding root element. In many cases, this implies that R and G(R) are bi-interpretable, yielding a new approach to bi-interpretability for algebraic groups over a wide range of rings and fields.For such groups it then follows that the group G(R) is (finitely) axiomatizable in the appropriate class of groups provided R is (finitely) axiomatizable in the corresponding class of rings.

Authors

I am an author on this paper

Click your name to claim this paper and add it to your profile.

Reviews

Primary Rating

4.5

Not enough ratings

Secondary Ratings

Novelty

-

Significance

-

Scientific rigor

-

Rate this paper

Recommended

Article Mathematics

A higher-dimensional Chevalley restriction theorem for orthogonal groups

Lei Song, Xiaopeng Xia, Jinxing Xu

Summary: We prove a higher-dimensional Chevalley restriction theorem for orthogonal groups, which confirms a conjecture by Chen and Ngo for reductive groups. Additionally, a weaker statement is proven for characteristic p > 2. The theorem implies the integral and normal nature of the categorical quotient of a commuting scheme by the diagonal adjoint action of the group in characteristic 0. Furthermore, we demonstrate trace identities and a certain multiplicative property of the Pfaffian over an arbitrary commutative algebra.

ADVANCES IN MATHEMATICS (2023)

Add to Collection

Article Multidisciplinary Sciences

Neural network interpretation using descrambler groups

Jake L. Amey, Jake Keeley, Tajwar Choudhury, Ilya Kuprov

Summary: Researchers have found a method for enhancing interpretability and trust in deep neural networks, successfully applied to a digital signal processing network in electron spin resonance. They discovered that the network is able to spontaneously conduct complex mathematical transformations and filtering operations internally without human intervention.

PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA (2021)

Add to Collection

Article Mathematics

Chevalley-Warning type results on abelian groups

Erhard Aichinger, Jakob Moosbauer

Summary: The study introduces a new concept of function degrees for abelian groups, allowing the generalization of Chevalley-Warning Theorems from fields to noncommutative rings or abelian groups of prime power order.

JOURNAL OF ALGEBRA (2021)

Add to Collection

Article Mathematics

NON-ASSOCIATIVE FROBENIUS ALGEBRAS FOR SIMPLY LACED CHEVALLEY GROUPS

Tom De Medts, Michiel Van Couwenberghe

Summary: In this study, a construction for a class of commutative, non-associative algebras for simple Chevalley groups of simply laced type is presented. Furthermore, these algebras are equipped with an associating bilinear form, converting them into Frobenius algebras. The class includes a 3876-dimensional algebra acted upon by the Chevalley group of type E-8, demonstrating the structure of (axial) decomposition algebras.

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY (2021)

Add to Collection

Article Mathematics

STRONG BOUNDEDNESS OF SPLIT CHEVALLEY GROUPS

Alexander A. Trost

Summary: This paper investigates the diameter of certain word norms on S-arithmetic split Chevalley groups and proves that the word metrics depend only on the number of conjugacy classes. It also provides examples of normal generating sets and offers a complete account of the existence of small normally generating sets for Sp(4) (R) and G(2) (R).

ISRAEL JOURNAL OF MATHEMATICS (2022)

Add to Collection

Article Mathematics

ON THE REMAINDER TERM OF THE WEYL LAW FOR CONGRUENCE SUBGROUPS OF CHEVALLEY GROUPS

Tobias Finis, Erez Lapid

Summary: This paper discusses the Weyl law for a locally symmetric space X defined by a simple Chevalley group G and a congruence subgroup of G(Q), with a power-saving estimate for the remainder term when G is simply connected, advancing the results of Lindenstrauss and Venkatesh.

DUKE MATHEMATICAL JOURNAL (2021)

Add to Collection

Article Mathematics

Bi-exactness of relatively hyperbolic groups

Koichi Oyakawa

Summary: We prove that finitely generated relatively hyperbolic groups are bi-exact if and only if all peripheral subgroups are bi-exact. This is a generalization of Ozawa's result which claims that finitely generated relatively hyperbolic groups are bi-exact if all peripheral subgroups are amenable.

JOURNAL OF FUNCTIONAL ANALYSIS (2023)

Add to Collection

Article Mathematics

Groups GL(∞) over finite fields and multiplications of double cosets

Yury A. Neretin

Summary: This paper focuses on the unitary representation theory of continuous linear operators group (GL) over bar over a finite field, reducing it to projective representations in a certain category and studying the application of linear relations in finite-dimensional linear spaces.

JOURNAL OF ALGEBRA (2021)

Add to Collection

Article Chemistry, Physical

Probabilistic Interpretation of NMR J-Couplings Determines BI-BII State Equilibria in DNA

Jiri Fukal, Marie Zgarbova, Petr Jurecka, Jakub Sebera, Vladimir Sychrovsky

Summary: The interpretation of 3JP,H3' NMR scalar spin-spin coupling constants in DNA can be improved by considering different structural states and using weighted-static or adiabatic-MD methods. Ad-MD calculations allow for cross-validation of different force fields and parameterizations and provide comparable results to experimental values.

JOURNAL OF CHEMICAL THEORY AND COMPUTATION (2022)

Add to Collection

Article Mathematics, Applied

Perfect state transfer on weighted bi-Cayley graphs over abelian groups

Shixin Wang, Tao Feng

Summary: This paper establishes necessary and sufficient conditions for a weighted bi-Cayley graph having perfect state transfer over any given finite abelian group. As special cases of weighted bi-Cayley graphs, the results in this paper can apply to examine weighted Cayley graphs having perfect state transfer over dihedral groups, generalized dihedral groups, semi-dihedral groups and dicyclic groups.

APPLIED MATHEMATICS AND COMPUTATION (2023)

Add to Collection

Article Mathematics, Applied

On cubic semisymmetric bi-Cayley graphs on nonabelian simple groups

Jiangmin Pan, Yingnan Zhang

Summary: In this paper, the authors classify cubic semisymmetric bi-Cayley graphs on nonabelian simple groups, which extends the classification of cubic nonnormal symmetric Cayley graphs on nonabelian simple groups.

AIMS MATHEMATICS (2022)

Add to Collection

Article Mathematics

Twisted conjugacy in classical Chevalley groups over certain domains of positive characteristic

Shripad M. Garge, Oorna Mitra

Summary: This article studies the relationship between subfields and rings in the algebraic closure of a finite field, and proves that a certain type of classical Chevalley adjoint group has the R-infinity property.

JOURNAL OF ALGEBRA (2022)

Add to Collection

Article Mathematics, Applied

Twisted conjugacy classes in twisted Chevalley groups

Sushil Bhunia, Pinka Dey, Amit Roy

Summary: This paper proves that twisted Chevalley groups over a field k of characteristic zero have the R-infinity-property, and they also have the S-infinity-property when the transcendence degree of k over Q is finite or Aut(k) is periodic.

JOURNAL OF ALGEBRA AND ITS APPLICATIONS (2022)

Add to Collection

Article Chemistry, Inorganic & Nuclear

Crystal-chemical and morphological interpretation of the biocompatibility of compounds in a Ca-Na-Bi-fluorapatite system

Evgeny N. Bulanov, Ksenia S. Stasenko, Oxana N. Golitsyna, Marpha N. Egorikhina, Diana Ya Aleynik, Nikolai E. Skoblikow, Alexander Knyazev

Summary: This study investigated the isomorphism features in compounds of a Ca-Na-Bi-P-O-F system with a crystalline structure of mineral apatite, and their effects on biocompatibility with human cells in an in vitro model. An anomaly of increased relative cell growth rate was observed for certain compounds, with discussions on possible crystal-chemical and morphological reasons for this phenomenon.

DALTON TRANSACTIONS (2022)

Add to Collection

Article Mathematics

On the A1-Invariance of K2 Modeled on Linear and Even Orthogonal Groups

Andrei Lavrenov, Sergey Sinchuk, Egor Voronetsky

Summary: In this paper, we prove the A(1)-invariance property of the unstable functor K2 in the linear and even orthogonal cases. This result has important implications for the representation of the fundamental groups of Chevalley-Demazure group schemes.

INTERNATIONAL MATHEMATICS RESEARCH NOTICES (2022)

Add to Collection

Biographical-Item Mathematics

Fritz Grunewald, 1949-2010 OBITUARY

Dan Segal

BULLETIN OF THE LONDON MATHEMATICAL SOCIETY (2012)

Add to Collection

Article Mathematics

Powers in finite groups

Nikolay Nikolov, Dan Segal

GROUPS GEOMETRY AND DYNAMICS (2011)

Add to Collection

Article Mathematics

Generators and commutators in finite groups; abstract quotients of compact groups

Nikolay Nikolov, Dan Segal

INVENTIONES MATHEMATICAE (2012)

Add to Collection

Article Mathematics

On verbal subgroups of adelic groups

Dan Segal

JOURNAL OF ALGEBRA (2011)

Add to Collection

Article Mathematics

Algorithms determining finite simple images of finitely presented groups

Martin R. Bridson, David M. Evans, Martin W. Liebeck, Dan Segal

INVENTIONES MATHEMATICAE (2019)

Add to Collection

Article Mathematics

Finite axiomatizability for profinite groups

Andre Nies, Dan Segal, Katrin Tent

Summary: In this paper, we investigate finitely axiomatizable groups in various classes, demonstrating that profinite groups and pro-p groups are finitely axiomatizable within their respective categories. Algebraic and model-theoretic methods are employed for this purpose, while reasons for certain groups not being finitely axiomatizable are also discussed.

PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY (2021)

Add to Collection

Article Mathematics

A profinite analogue of Lasserre's theorem

Dan Segal

Summary: We characterize the virtually soluble profinite groups of finite rank that are finitely axiomatizable in the class of all profinite groups.

JOURNAL OF ALGEBRA (2023)

Add to Collection

Article Mathematics

Remarks on profinite groups having few open subgroups

Dan Segal

JOURNAL OF COMBINATORIAL ALGEBRA (2018)

Add to Collection

Article Mathematics

Finitely generated groups with polynomial index growth

Laszlo Pyber, Dan Segal

JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK (2007)

Add to Collection

Article Mathematics

Breadth in polycyclic groups

Avinoam Mann, Dan Segal

INTERNATIONAL JOURNAL OF ALGEBRA AND COMPUTATION (2007)

Add to Collection

Article Mathematics

A characterization of finite soluble groups

Nikolay Nikolov, Dan Segal

BULLETIN OF THE LONDON MATHEMATICAL SOCIETY (2007)

Add to Collection

Article Mathematics

On the group rings of abelian minimax groups, II: The singular case

Dan Segal

JOURNAL OF ALGEBRA (2006)

Add to Collection

Article Mathematics

On the integer solutions of quadratic equations

F Grunewald, D Segal

JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK (2004)

Add to Collection

Article Mathematics

Finite index subgroups in profinite groups

N Nikolov, D Segal

COMPTES RENDUS MATHEMATIQUE (2003)

Add to Collection

No Data Available

© Peeref 2019-2024. All rights reserved.