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## Publications

Publications (192)

Let R be an associative ring with 1 and G = GL(n, R) the general linear group of degree n ≥ 3 over R. A goal of the paper is to calculate the relative centralizers of the relative elementary subgroups or the principal congruence subgroups, corresponding to an ideal A ⊴ R modulo the relative elementary subgroups or the principal congruence subgroups...

We prove that Chevalley groups over polynomial rings $\mathbb F_q[t]$ and over Laurent polynomial $\mathbb F_q[t,t^{-1}]$ rings, where $\mathbb F_q$ is a finite field, are boundedly elementarily generated. Using this we produce explicit bounds of the commutator width of these groups. Under some additional assumptions, we prove similar results for o...

Let R be any associative ring with 1, n ≥ 3, and let A, B be two-sided ideals of R. In our previous joint works with Roozbeh Hazrat [17], [15], we have found a generating set for the mixed commutator subgroup [E(n, R, A); E(n, R, B)]. Later in [29], [34] we noticed that our previous results can be drastically improved and that [E(n, R, A); E(n, R,...

Recently, Raimund Preusser displayed very short polynomial expressions of elementary generators in classical groups over an arbitrary commutative ring as products of conjugates of an arbitrary matrix and its inverse by absolute elementary matrices. In particular, this provides very short proofs for description of normal subgroups. In 2018, the auth...

Let R be any associative ring with 1, let n ≥ 3, and let A,B be two-sided ideals of R. In the present paper, we show that the mixed commutator subgroup [E(n,R,A),E(n,R,B)] is generated as a group by the elements of the two following forms: 1) zij(ab, c) and zij (ba, c), 2) [tij(a), tji(b)], where 1 ≤ i ≠ j ≤ n, a ∈ A, b ∈ B, c ∈ R. Moreover, for th...

In this part I discuss the role of computers in the current research on the additive number theory, in particular in the solution of the classical Waring problem. In its original XVIII century form this problem consisted in finding for each natural k the smallest such s=g(k) that all natural numbers n can be written as sums of s non-negative k-th p...

In a joint paper of the author with Alexei Stepanov, it was established that for any two comaximal ideals A and B of a commutative ring R, A + B = R, and any n ≥ 3 one has [E(n,R,A),E(n,R,B)] = E(n,R,AB). Alec Mason and Wilson Stothers constructed counterexamples demonstrating that the above equality may fail when A and B are not comaximal, even fo...

In the last decades there was much ado about computer proofs, computer aided proofs, computer verified proofs, etc. It is obvious that the advent and proliferation of computers have drastically changed applications of mathematics. What one discusses much less, however, is how computers changed mathematics itself, and mathematicians’ stance in regar...

Let $R$ be an associative ring with 1, $G=GL(n, R)$ be the general linear group of degree $n\ge 3$ over $R$. In this paper we calculate the relative centralisers of the relative elementary subgroups or the principal congruence subgroups, corresponding to an ideal $A\unlhd R$ modulo the relative elementary subgroups or the principal congruence subgr...

Let $R$ be any associative ring with $1$, $n\ge 3$, and let $A,B$ be two-sided ideals of $R$. In our previous joint works with Roozbeh Hazrat [17,15] we have found a generating set for the mixed commutator subgroup $[E(n,R,A),E(n,R,B)]$. Later in [29,34] we noticed that our previous results can be drastically improved and that $[E(n,R,A),E(n,R,B)]$...

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 Bak's unitary groups. It is a direct sequel of our similar results were obtained for $GL(n,R)$ and for Chevalley groups over a commutative ring with 1, respectively. Namel...

In our previous joint papers with Roozbeh Hazrat and Alexei Stepanov we established commutator formulas for relative elementary subgroups in GL(n,R), n≥3, and other similar groups, such as Bak's unitary groups, or Chevalley groups. In particular, there it was shown that multiple commutators of elementary subgroups can be reduced to double such comm...

Nowhere in mathematics is the progress resulting from the advent of computers is as apparent, as in the additive number theory. In this part, we describe the role of computers in the investigation of the oldest function studied in mathematics, the divisor sum. The disciples of Pythagoras started to systematically explore its behaviour more that 250...

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 relative elementary subgroups in Chevalley groups. Namely, let $\Phi$ be a reduced irreducible root system of rank $\ge 2$, let $R$ be a commutative ring and let $A,B$ b...

In the present paper, which is a direct sequel of our paper [14] joint with Roozbeh Hazrat, we prove an unrelativized version of the standard commutator formula in the setting of Chevalley groups. Namely, let Φ be a reduced irreducible root system of rank ≥ 2, let R be a commutative ring and let I , J be two ideals of R . We consider subgroups of t...

Decomposition of unipotents gives short polynomial expressions of the conjugates of elementary generators as products of elementaries. It turns out that with some minor twist the decomposition of unipotents can be read backwards to give very short polynomial expressions of the elementary generators themselves in terms of elementary conjugates of an...

In the present note, which is a marginalia to the previous papers by Roozbeh Hazrat, Alexei Stepanov, Zuhong Zhang, and the author, I observe that for any ideals A,B≤R of a commutative ring R and all n ≥ 3 the birelative standard commutator formula also holds in the unrelativized form, as [E(n,A),GL(n,B)] = [E(n,A),E(n,B)] and discuss some obvious...

In the present paper we continue the study of the elementary commutator subgroups $[E(n,A),E(n,B)]$, where $A$ and $B$ are two-sided ideals of an associative ring $R$, $n\ge 3$. First, we refine and expand a number of the auxiliary results, both classical ones, due to Bass, Stein, Mason, Stothers, Tits, Vaserstein, van der Kallen, Stepanov, as also...

In our previous joint papers with Roozbeh Hazrat and Alexei Stepanov we established commutator formulas for relative elementary subgroups in $GL(n,R)$, $n\ge 3$, and other similar groups, such as Bak's unitary groups, or Chevalley groups. In particular, there it was shown that multiple commutators of elementary subgroups can be reduced to double su...

Let $R$ be any associative ring with $1$, $n\ge 3$, and let $A,B$ be two-sided ideals of $R$. In the present paper we show that the mixed commutator subgroup $[E(n,R,A),E(n,R,B)]$ is generated as a group by the elements of the two following forms: 1) $z_{ij}(ab,c)$ and $z_{ij}(ba,c)$, 2) $[t_{ij}(a),t_{ji}(b)]$, where $1\le i\neq j\le n$, $a\in A$,...

The simplistic view of Mathematics as a logical system of formal truths deduced from a limited set of axioms by a limited set of inference rules immediately shatters when confronted with the history of Mathematics, or current mathematical practice. To become useful, mathematical Philosophy should contemplate what Mathematics actually was, over cent...

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 Chevalley groups. Namely, let $\Phi$ be a reduced irreducible root system of rank $\ge 2$, let $R$ be a commutative ring and let $I,J$ be two ideals of $R$. We consider subgr...

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 group. Namely, he proved that for an associative ring A
with identity, E(A)=[E(A),E(A)]=[GL(A),GL(A)] where GL(A) is the stable general
linear group and E(A) is its elementa...

In this paper I sketch two new variations of the method of decomposition of unipotents in the microweight representations (E6,ϖ1) and (E7,ϖ7). To put them in the context, I first very briefly recall the two previous stages of the method, an A5-proof for E6 and an A7-proof for E7, first developed some 25 years ago by Alexei Stepanov, Eugene Plotkin,...

The simply connected Chevalley group G(E7, R) of type E7 is considered in the 56-dimensional representation. The main objective is to prove that the following four groups coincide: the normalizer of the elementary Chevalley group E(E7, R), the normalizer of the Chevalley group G(E7, R) itself, the transporter of E(E7, R) into G(E7, R), and the exte...

This paper is a slightly expanded text of our talk at the PCA-2014. There we announced two recent results concerning explicit polynomial equations defining exceptional Chevalley groups in microweight or adjoint representations. One of these results is an explicit characteristic-free description of equations on the entries of a matrix from the simpl...

In the present paper, we describe some recent applications of localization methods to the study of commutators in the groups of points of algebraic and algebraic-like groups, such as GL(n,R), Bak’s unitary groups GU(2n,R, Λ), and Chevalley groups G(Φ,R). In particular, we announce the multiple relative commutator formula and the general multiple re...

We revisit localisation and patching method in the setting of Chevalley
groups. Introducing certain subgroups of relative elementary Chevalley groups,
we develop relative versions of the conjugation calculus and the commutator
calculus in Chevalley groups $G(\Phi,R)$, $\rk(\Phi)\geq 2$, which are both
more general, and substantially easier than the...

An embedding of root systems ∆ ⊆ Φ determines the corresponding regular embedding G(∆, R)≤ G(Φ, R) of Chevalley groups, over an arbitrary commutative ring R. Denote by E(∆, R) the elementary subgroup of G(∆, R). In the present paper we initiate the study of intermediate subgroups H, E∆, R) ≤ H ≤ G(Φ, R), provided that Φ=E6, E7, E8, F4 or G2, and th...

Let R be a commutative ring all of whose proper factor rings are finite and such that there exists a unit of infinite order. We show that for a subgroup P in G = SO(2l, R), l ≥ 3, containing the Borel subgroup B, the following alternative holds: either P contains a relative elementary subgroup E
I for some ideal I = 0 or H is contained in a proper...

In this paper we study some remarkable semisimple elements of an (extended) Chevalley group that are diagonalizable over the ground field – the ‘weight elements’. In particular, we calculate the Bruhat decomposition of microweight elements. Results of the present paper are crucial for the description of overgroups of split maximal tori in Chevalley...

Recently Liebeck, Nikolov, and Shalev noticed that finite Chevalley groups admit fundamental SL2-factorizations of length 5N, where N is the number of positive roots. From a recent paper by Smolensky, Sury, and Vavilov, it follows that the elementary Chevalley groups over rings of stable rank 1 admit such factorizations of length 4N. In the present...

This paper is the first part of a systematic survey on the structure of classical groups over general rings. We intend to cover various proofs of the main structure theorems, commutator formulas, finiteness and stability conditions, stability and prestability theorems, the nilpotency of K
1, the centrality of K
2, automorphisms and homomorphisms, e...

Let $\Phi$ be a reduced irreducible root system of rank $\ge 2$, let $R$ be a
commutative ring and let $I,J$ be two ideals of $R$. In the present paper we
describe generators of the commutator groups of relative elementary subgroups
$\big[E(\Phi,R,I),E(\Phi,R,J)\big]$ both as normal subgroups of the elementary
Chevalley group $E(\Phi,R)$, and as gr...

The author and M. R. Gavrilovich [in part I, St. Petersbg. Math. J. 16, No. 4, 649-672 (2005); translation from Algebra Anal. 16, No. 4, 54-87 (2004; Zbl 1105.20039)] proposed a geometric proof of the structure theorems for Chevalley groups of types Φ=E 6 ,E 7 , based on the following fact. There are nontrivial root unipotents of type A 2 stabilizi...

The study of subgroups H such that E(m,R)⊗E(n,R)≤H≤G= GL (mn,R) is started, provided that the ring R is commutative and m,n≥3. The principal results of this part can be summarized as follows. The group GL m ⊗ GL n is described by equations, and it is proved that the elementary subgroup E(m,R)⊗E(n,R) is normal in ( GL m ⊗ GL n )(R). Moreover, when m...

An analog of the Dennis-Vaserstein decomposition is proved for an arbitrary pair of maximal parabolic subgroups Pr and Ps in split classical groups, under appropriate stability conditions. Before, such decompositions were only known for pairs of terminal parabolic subgroups.

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 that [elementary] Chevalley groups very rarely
have finite commutator width. The reason is that they have very few
commutators, in fact, commutators have finite width in e...

Let $(\FormR)$ be a form ring such that $A$ is quasi-finite $R$-algebra
(i.e., a direct limit of module finite algebras) with identity. We consider the
hyperbolic Bak's unitary groups $\GU(2n,\FormR)$, $n\ge 3$. For a form ideal
$(I,\Gamma)$ of the form ring $(\FormR)$ we denote by $\EU(2n,I,\Gamma)$ and
$\GU(2n,I,\Gamma)$ the relative elementary g...

It is shown that the problem of describing those subgroups in the general linear group GL(n, R) which are normalized by a classical group is much more difficult than believed previously. For the case of even orthogonal groups, a thorough level calculation is performed, which shows that, even under the assumption 2 ∈ R*, the level of a subgroup H ≤...

This note revisits localisation and patching method in the setting of
generalised unitary groups. Introducing certain subgroups of relative
elementary unitary groups, we develop relative versions of the conjugation
calculus and the commutator calculus in unitary groups, which are both more
general, and substantially easier than the ones available i...

Let G = G(Φ,R) be the simply connected Chevalley group with root system Φ over a ring R. Denote by E(Φ,R) its elementary subgroup. The main result of the article asserts that the set of commutators C = {[a, b]|a ∈ G(Φ, R), b ∈ E(Φ, R)} has bounded width with respect to elementary generators. More precisely, there exists a constant L depending on Φ...

In the 1960's Noboru Iwahori and Hideya Matsumoto, Eiichi Abe and Kazuo
Suzuki, and Michael Stein discovered that Chevalley groups $G=G(\Phi,R)$ over a
semilocal ring admit remarkable Gauss decomposition $G=TUU^-U$, where
$T=T(\Phi,R)$ is a split maximal torus, whereas $U=U(\Phi,R)$ and
$U^-=U^-(\Phi,R)$ are unipotent radicals of two opposite Borel...

Lately, the following problem has attracted a lot of attention in various
contexts: find the shortest factorisation $G=UU^-UU^-...U^{\pm}$ of a Chevalley
group $G=G(\Phi,R)$ in terms of the unipotent radical $U=U(\Phi,R)$ of the
standard Borel subgroup $B=B(\Phi,R)$ and the unipotent radical
$U^-=U^-(\Phi,R)$ of the opposite Borel subgroup $B^-=B^-...

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 describe relative localisation, universal localisation, and
enhanced versions of localisation-completion. Apart from the general strategic
description of these methods, we stat...

The present paper is devoted to a detailed computer study of the action of the Chevalley group G(E
6, R) on the minimal module V(ῶ
1). Our main objectives are an explicit choice and a tabulation of the signs of structure constants for this action, compatible
with the choice of a positive Chevalley base, the construction of multilinear invariants an...

The method of decomposition of unipotents consists in writing elementary matrices as products of factors lying in proper parabolic
subgroups whose images under inner automorphisms also fall into proper parabolic subgroups of certain types. For the general
linear group, this method was first proposed by Stepanov in 1987 to simplify the proof of Susl...

The paper deals with some additional details concerning the parametrization of the highest Weyl orbit of equations on the
highest weight orbit in the adjoint representations of Chevalley groups of types E7 and E8, as given in the author’s paper “Numerology of square equations.” Bibliography: 25 titles.

A generalization of the Dennis–Vaserstein decomposition is proved for an arbitrary pair of maximal parabolic subgroups P
r
and P
s
in the general linear group GL(n, R), provided that r − s ≥ sr (R). The usual Dennis–Vaserstein decomposition is the special case where r = n − 1, s =1. Bibliography: 23 titles.

Let R be a commutative ring all of whose proper factor rings are finite and in which a unit of infinite order exists. It is shown that for a subgroup P in G =SL(n, R), n ≥3,or in G =Sp(2l, R), l ≥2, containing the standard Borel subgroup B, the following alternative holds: either P contains a relative elementary subgroup EI for some ideal I≠0 or H...

In the present paper, we discuss a major project whose goal is to develop theoretical background and working algorithms for calculations in exceptional Chevalley groups over commutative rings. We recall some basic facts concerning calculations in groups over fields, and indicate complications arising in the ring case. Elementary calculations as suc...

We finish the proof of the main structure theorems for a Chevalley group G(Φ, R) of rank ≥ 2 over an arbitrary commutative ring R. Namely, we prove that for any admissible pair (A, B) in the sense of Abe, the corresponding relative elementary group E(Φ,R, A, B) and the full congruence subgroup C(Φ, R, A, B) are normal in G(Φ, R) itself, and not jus...

Let R be an associative ring with 1; A, B ⊴ R be its ideals; C(n, R, A) be the full congruence subgroup of level A in GL(n, R); and E(n, R, A) be the relative elementary subgroup of level A. We present a very easy proof of the following commutator formula: [E(n, R, A),C(n, R, B)] = [E(n, R, A), E(n, R, B)] for all commutative rings based exclusivel...

We prove here some supplementary statements that appeared without proof in I. Panin, A. Stavrova, N. Vavilov, On Grothendieck--Serre's conjecture concerning principal $G$-bundles over reductive group schemes:I, arXiv:0905.1418 Comment: We prove some supplementary statements that appeared without proof in arXiv:0905.1418; 10 pages

In the present work, which is a sequel of the paper "Can one see the signs of structure constants?", we describe how one can see the form and the signs of the senior Weyl orbit of equations on the highest weight orbit directly in the weight diagram of microweight representations and adjoint representations for the simply-laced case. As special case...

The subgroups E(m,R) ⊗ E(n,R) ≤ H ≤ G = GL(mn,R) are studied under the assumption that the ring R is commutative and m, n ≥ 3. The group GL
m
⊗GL
n
is defined by equations, the normalizer of the group E(m,R) ⊗ E(n,R) is calculated, and with each intermediate subgroup H it is associated a uniquely determined lower level (A,B,C), where A,B,C are idea...

The method of decomposition of unipotents consists of writing elementary matrices as products of factors lying in proper parabolic
subgroups, whose images under (abstract) inner automorphisms also fall into proper parabolic subgroups of certain types. For
the general linear group, this method was first proposed by Stepanov in 1987 to simplify the p...

This paper studies the work of Bak in Algebra and (lower) Algebraic K- theory and some later developments stimulated by them. We present an overview of his work in these areas, describe the setup and problems as well as the methods he introduced to attack these problems and state some of the crucial theorems. The aim is to analyse in detail some of...

In the present work, which is a sequel of [St. Petersbg. Math. J. 19, No. 4, 519-543 (2008), translation from Algebra Anal. 19, No. 4, 34-68 (2007; Zbl 1203.20041)], we describe how one can see the form and the signs of the senior Weyl orbit of equations on the highest weight orbit directly in the weight diagram of microweight representations and a...

Let G and E stand for one of the following pairs of groups:• Either G is the general quadratic group U(2n,R,Λ), n≥3, and E its elementary subgroup , for an almost commutative form ring (R,Λ),• or G is the Chevalley group G(Φ,R) of type Φ, and E its elementary subgroup E(Φ,R), where Φ is a reduced irreducible root system of rank ≥2 and R is commutat...

In the present paper we prove the main structure theorem for Chevalley groups G=G(Φ;R) of types Φ=E 6 ,E 7 over a commutative ring R. More precisely, we describe subgroups in G normalized by the elementary subgroup E(Φ;R). This result is not new, since structure theorems are known for all Chevalley groups (see the bibliography for references). The...

Let k be an infinite field. Let R be the semi-local ring of a finite family
of closed points on a k-smooth affine irreducible variety, let K be the
fraction field of R, and let G be a reductive simple simply connected R-group
scheme isotropic over R. We prove that for any Noetherian k-algebra A, the map
of etale cohomology sets H^1(A\otimes_k R,G)-...

The unipotent decomposition method consists in representing elementary matrices as products of factors belonging to proper parabolic subgroups whose images under endomorphisms (e.g., conjugations) remain in proper parabolic subgroup. For the complete linear group, this method was suggested in 1987 by Stepanov, who applied it to simplify the proof o...

We consider the simply connected Chevalley group $G(E_7,R)$ of type $E_7$ in the 56-dimensional representation. The main objective of the paper is to prove that the following four groups coincide: the normalizer of the elementary Chevalley group $E(E_7,R)$, the normalizer of the Chevalley group $G(E_7,R)$ itself, the transporter of $E(E_7,R)$ into...

It is described how one can see the signs of structure constants of an action directly in the weight diagram of microweight and adjoint representations for groups of types E 6 , E 7 , and E 8 . This generalizes the results of the preceding paper, [Rend. Semin. Mat. Univ. Padova 104, 201-250 (2000; Zbl 1016.20029)], where a similar algorithm was dis...

Let Γ = GSp(2l, R) be the general symplectic group of rank l over a commutative ring R such that 2 ∈ R*; and let ν be a symmetric equivalence
relation on the index set {1,…, l, −l,…, 1} all of whose classes contain at least 3 elements. In the present paper, we prove
that if a subgroup H of Γ contains the group EΓ(ν) of elementary block diagonal mat...

Classification of subgroups in a Chevalley group G(Φ, R) over a commutative ring R, normalized by the elementary subgroup E(Φ, R), is well known. However, for exceptional groups, in the available literature neither the parabolic reduction nor the level reduction can be found. This is due to the fact that the Abe-Suzuki-Vaserstein proof relied on lo...

We describe the orbits of the general linear group GL(n,T) over a skew field T acting by simultaneous conjugation on pairs of 1-tori, i.e., subgroups conjugate to diag(T * ,1,⋯,1), and identify the corresponding spans. We also provide some applications of these results to the description of intermediate subgroups and generation. These results were...

Let R be a commutative ring with 1, n a natural number, and let l = [n/2]. Suppose that 2 E R* and l >= 3. We describe the subgroups of the general linear group GL(n, R) that contain the elementary orthogonal group EO(n, R). The main result of the paper says that, for every intermediate subgroup H, there exists a largest ideal A <= R such that EEO(...

Let R be a commutative ring with 1, A, B ⊴ R be its ideals, GL(n, R, A) be the principal congruence subgroup of level A in GL(n, A), and E(n, R, A) be the relative elementary subgroup of level A. We prove the following commutator formula
$$
[E(n,R,A),GL(n,R,B)] = [E(n,R,A),E(n,R,B)],
$$
which generalizes known results. The proof is yet another v...

The paper is devoted to a detailed study of some remarkable semisimple elements of (extended) Chevalley groups that are diagonalizable over the ground field – the weight elements. These are the conjugates of certain semisimple elements h ω (ε) of extended Chevalley groups G ¯=G ¯(Φ,K), where ω is a weight of the dual root system Φ ∨ and ε∈K * . In...

In the present paper, it is proved that if R is a commutative semilocal ring all the residue fields of which contain at least
3n + 2 elements, then for every subgroup H of the special linear group SL(n, R), n ≥ 3, containing the diagonal subgroup SD(n, R) there exists a unique D-net σ of ideals of R such that Γ(σ)≤H≤NΓ(σ). In works by Z. I. Borewic...

Let G = G(Φ, K) be a Chevalley group over a field K of characteristic ≠ 2. In the present paper, we classify the subgroups
of G generated by triples of long root subgroups, two of which are opposite, up to conjugacy. For finite fields, this result
is contained in papers by B. Cooperstein on the geometry of root subgroups, whereas for SL (n, K) it i...

In the present paper, we characterize ⋀n(GL(n, R)) over any commutative ring R as the connected component of the stabilizer of the Plücker ideal. This folk theorem is
classically known for algebraically closed fields and should also be well known in general. However, we are not aware of any
obvious reference, so we produce a detailed proof, which f...

Let Φ be a reduced irreducible root system. We consider pairs (S, X (S)), where S is a closed set of roots, X(S) is its stabilizer
in the Weyl group W(Φ). We are interested in such pairs maximal with respect to the following order: (S1, X (S1)) ≤ (S2, X (S2)) if S1 ⊆ S2 and X(S1) ≤ X(S2). The main theorem asserts that if Δ is a root subsystem such...

In the nineties the author, A. Stepanov and E. Plotkin developed a geometric approach towards calculations in exceptional groups at the level of K1, decomposition of unipotents. However, it relied on the presence of large classical embeddings, such as A5 ≤ E6 or A7 ≤ E7. Recently the author, M. Gavrilovich and S. Nikolenko devised a sharper geometr...

In this paper we prove that the groupws SL(n,q), q=pm, are factors of the modular groups PSL(2,Z) When n=5,6,7 and P≠2, q≠9

Different geometric proofs of the main structure theorems for Chevalley groups over commutative rings are described and compared.
Known geometric proofs, published by I. Z. Golubchik, N. A. Vavilov, A. V. Stepanov, and E. B. Plotkin, such as A2 and A3 proofs for classical groups, A5 and D5 proofs for E6, A7 and D6 proofs for E7, and a D8 proof for...

In the present paper we prove the main structure theorem for Chevalley groups G = G(Φ, R) of types Φ = E6, E7 over a commutative ring R. More precisely, we describe subgroups in G normalized by the elementary subgroup E(Φ, R). This result is not new, since structure theorems are known for all Chevalley groups [25, 27, 28, 30], [38]–[40], and [58, 6...

Let R be an associative ring with 1, n. We show that Higman''s computation of the first cohomology group of the special linear group over a field with natural coefficients actually demonstrates that H
1(St)(n,R),R
n=for n4, and we explicitly compute this group for n=3, when it does not vanish. The second-named author extended these results to all c...

In the first paper of the series, we proved the standardness of a subgroup H containing a split maximal torus in the split spinor group Spin(n,R) over a field K of characteristic different from 2 containing at least 7 elements under one of the following additional assumptions: (1) H is reducible, (2) H is imprimitive, (3) H contains a nontrivial ro...

Let R be a commutative ring, and let l ‚ 2; for l = 2 it is assumed additionally that R has no residue fields of two elements. The subgroups of the general linear group GL(n;R) that contain the elementary symplectic group Ep(2l;R) are described. In the case where R = K is a field, similar results were obtained earlier by Dye, King, and Shang Zhi Li...

Two algorithms for computing the structure table of Lie algebras of type E
l
with respect to a Chevalley base are compared: the usual inductive algorithm and an algorithm based on the use of the Frenkel-Kac cocycle. It turns out that the Frenkel–Kac algorithm is several dozen times faster, but under the natural choice of a bilinear form and a sign...

The subgroups of the split orthogonal group = GO(n,R) of degree n over a commutative ring R with 2 R* that contain the elementary subgroup of a regularly embedded semisimple group F are described. It is shown that if the ranks of all irreducible components of F are at least 4, then the description of its overgroups is standard in the sense that for...

Let R be a commutative ring with 1, let 2 ∈ R
*, and let l ≥ 3. We describe the subgroups of the general linear group GL(n,R) that contain the split elementary orthogonal group EO(2l,R). For every intermediate subgroup H, there exists a unique maximal ideal A ⊴ R such that E(2l,R,A) ≤ H and, moreover, H normalizes EO(2l,R)E(2l,R,A). In the case whe...

Let Φ be a reduced irreducible root system and R be a commutative ring. Further, let G(Φ,R) be a Chevalley group of type Φ over R and E(Φ,R) be its elementary subgroup. We prove that if the rank of Φ is at least 2 and the Bass-Serre dimension of R is finite, then the quotient G(Φ,R)/E(Φ,R) is nilpotent by abelian. In particular, when G(Φ,R) is simp...

In the first paper of the present series, we proved the standardness of subgroups containing a split maximal torus in the split orthogonal group SO(n,R) over a commutative semilocal ring R for the following two situations: (1) n is even; (2) n is odd and R = K is a field. In the present paper, we prove the standardness of intermediate subgroups ove...