30 citations · 90 across the 20 of their papers we have counts for
20 papers
Relative centralisers of relative subgroups
Nikolai Vavilov, Zuhong Zhang
Let be an associative ring with 1, be the general linear group of degree over . In this paper we calculate the relative centralisers of the relative el…
Commutators of elementary subgroups: curiouser and curiouser
Nikolai Vavilov, Zuhong Zhang
Let be any associative ring with , , and let be two-sided ideals of . In our previous joint works with Roozbeh Hazrat [17,15] we have found a generating set…
Commutators of relative and unrelative elementary unitary groups
Nikolai Vavilov, Zuhong Zhang
In the present paper we find generators of the mixed commutator subgroups of relative elementary groups and obtain unrelativised versions of commutator formulas in the setting of B…
Commutators of relative and unrelative elementary subgroups in Chevalley groups
Nikolai Vavilov, Zuhong Zhang
In the present paper, which is a direct sequel of our papers [10,11,35] joint with Roozbeh Hazrat, we achieve a further dramatic reduction of the generating sets for commutators of…
Inclusion among commutators of elementary subgroups
Nikolai Vavilov, Zuhong Zhang
In the present paper we continue the study of the elementary commutator subgroups , where and are two-sided ideals of an associative ring , . Fi…
Multiple commutators of elementary subgroups: end of the line
Nikolai Vavilov, Zuhong Zhang
In our previous joint papers with Roozbeh Hazrat and Alexei Stepanov we established commutator formulas for relative elementary subgroups in , , and other similar…
Relative and unrelative elementary groups, revisited
Nikolai Vavilov, Zuhong Zhang
Let be any associative ring with , , and let be two-sided ideals of . In the present paper we show that the mixed commutator subgroup …
An elementary description of without elementary matrices
Thomas Huettemann, Zuhong Zhang
Let be a ring with unit. Passing to the colimit with respect to the standard inclusions (which add a unit vector as new last row and column) yields, by…
Generation of relative commutator subgroups in Chevalley groups. II
Nikolai Vavilov, Zuhong Zhang
In the present paper, which is a direct sequel of our paper [12] joint with Roozbeh Hazrat, we prove unrelativised version of the standard commutator formula in the setting of Chev…
Jump loci for the rank of matrices and Betti numbers of chain complexes over Laurent polynomial rings
Thomas Huettemann, Zuhong Zhang
Let be a non-empty set of ideals of the commutative ring , closed under taking smaller ideals. A subset of the group ring is called a -set if the id…
The commutators of classical groups
R. Hazrat, N. Vavilov, Z. Zhang
In his seminal paper, half a century ago, Hyman Bass established a commutator formula in the setting of (stable) general linear group which was the key step in defining the K_1 gro…
Triangular objects and systematic K-theory
Thomas Huettemann, Zuhong Zhang
We investigate modules over "systematic" rings. Such rings are "almost graded" and have appeared under various names in the literature; they are special cases of the G-systems of G…
A matrix description for of graded rings
Zuhong Zhang
The current paper is dedicated to the study of the classical groups of graded rings. Let be a graded ring with identity , where the grading is an abelian group…
Generation of relative commutator subgroups in Chevalley groups
Roozbeh Hazrat, Nikolai Vavilov, Zuhong Zhang
Let be a reduced irreducible root system of rank , let be a commutative ring and let be two ideals of . In the present paper we describe generators of the c…
Commutator width in Chevalley groups
Roozbeh Hazrat, Alexei Stepanov, Nikolai Vavilov +1
The present paper is the [slightly expanded] text of our talk at the Conference "Advances in Group Theory and Applications" at Porto Cesareo in June 2011. Our main results assert t…
Multiple Commutator Formulas for Unitary Groups
Roozbeh Hazrat, Nikolai Vavilov, Zuhong Zhang
Let $(\FormR)$ be a form ring such that is quasi-finite -algebra (i.e., a direct limit of module finite algebras) with identity. We consider the hyperbolic Bak's unitary gro…
Multiple Commutator Formulas
R. Hazrat, Z. Zhang
Let A be a quasi-finite R-algebra (i.e., a direct limit of module finite algebras) with identity. Let I_i, i=0,...,m, be two-sided ideals of A, \GL_n(A,I_i) the principal congruenc…
Relative commutator calculus in Chevalley groups
Roozbeh Hazrat, Nikolai Vavilov, Zuhong Zhang
We revisit localisation and patching method in the setting of Chevalley groups. Introducing certain subgroups of relative elementary Chevalley groups, we develop relative versions…
The yoga of commutators
R. Hazrat, A. V. Stepanov, N. A. Vavilov +1
In the present paper we discuss some recent versions of localisation methods for calculations in the groups of points of algebraic-like and classical-like groups. Namely, we descri…
Relative unitary commutator calculus and applications
R. Hazrat, N. Vavilov, Z. Zhang
This note revisits localisation and patching method in the setting of generalised unitary groups. Introducing certain subgroups of relative elementary unitary groups, we develop re…