K1 of Chevalley groups are nilpotent

Hazrat, Roozbeh; Vavilov, N. A.

doi:10.1016/s0022-4049(02)00292-x

Cited by 84 publications

(108 citation statements)

References 22 publications

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“…However, the main result of the paper [71] by Giovanni Taddei, which plays a crucial role in the proof of our Theorem 2, asserts that for rk(Φ) ≥ 2 this is not the case. For classical groups, similar results had been proved earlier by Andrei Suslin and Viacheslav Kopeiko; see [33,34,51,70,73] for the history of this result, other proofs and generalizations.…”

Section: N (E(e 6 R)) = N (G(e 6 R)) = Tran(e(e 6 R) G(e 6 Rsupporting

confidence: 77%

“…Moreover, in the papers [72] by Leonid Vaserstein and [51] by Roozbeh Hazrat and the first author [51], the following stronger results can be found, each implying, in particular , Lemma 1 in the above form. In the majority of existing constructions, the Chevalley group G = G(Φ, R) arises together with an action on a Weyl module V = V (ω), for a dominant weight ω. Denote by Λ = Λ(ω) the multiset of weights of the module V = V (ω) with multiplicity.…”

Section: Lemma 1 For Rk(φ) ≥ 2 the Elementary Subgroup E(φ R) Is Nmentioning

confidence: 88%

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Normalizer of the Chevalley group of type ${\mathrm E}_6$

Vavilov

Luzgarev²

2008

St. Petersburg Math. J.

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This is true over an arbitrary commutative ring R, all normalizers and transporters being taken in GL(27, R). Moreover, G(E 6 , R) is characterized as the stabilizer of a system of quadrics. This result is classically known over algebraically closed fields; in the paper it is established that the corresponding scheme over Z is smooth, which implies that the above characterization is valid over an arbitrary commutative ring. As an application of these results, we explicitly list equations a matrix g ∈ GL(27, R) must satisfy in order to belong to G(E 6 , R). These results are instrumental in a subsequent paper of the authors, where overgroups of exceptional groups in minimal representations will be studied.

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Section: N (E(e 6 R)) = N (G(e 6 R)) = Tran(e(e 6 R) G(e 6 Rsupporting

confidence: 77%

Section: Lemma 1 For Rk(φ) ≥ 2 the Elementary Subgroup E(φ R) Is Nmentioning

confidence: 88%

Normalizer of the Chevalley group of type ${\mathrm E}_6$

Vavilov

Luzgarev²

2008

St. Petersburg Math. J.

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“…Actually, the main result of [58] asserts exactly that for rk(Φ) ≥ 2 the elementary subgroup does not depend on the choice of T . For the classical groups this was established earlier by Suslin and Kopeiko [23,24,20]; see [15,33,34,43,56,65,67] for the history of this result, other proofs and generalizations.…”

Section: §3 Basic Notationmentioning

confidence: 65%

“…For exceptional groups, this result was first proved in [58] and [61] (in [15,43,65,67] different proofs can be found for the cases of Φ = E l relevant for the present paper; for classical groups compare also [3,9,34,44,45,56] and [60]- [64]). The relations…”

Section: Weyl Modulesmentioning

confidence: 71%

“…Namely, we invoke the fact that the standard commutator formulas are satisfied in G(Φ, R) for all ideals I R. In the case of exceptional groups, this result was first proved in full generality by Taddei and Vaserstein [58,61] (see also [15,65,67,43]). In principle, this theorem as well could be demonstrated by the methods of the present paper.…”

Section: Proposition 1 (Level Reduction) Assume That For Any Pair (Rmentioning

confidence: 99%

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An $\mathrm {A}_2$-proof of structure theorems for Chevalley groups of types $\mathrm {E}_6$ and $\mathrm {E}_7$

Vavilov

Gavrilovich

2005

St. Petersburg Math. J.

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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 [25,27,28,30], [38]-[40], and [58,61] (see [42,65,34,56] for further references). The gist of the present paper resides not in the results themselves, but rather in the method of their proof based on the geometry of exceptional groups. We believe that this method is novel and of significant interest. Actually the Schwerpunkt of the present paper abides in a new descent procedure, which enables reduction to groups of smaller rank. This procedure is both simpler and more powerful than any other method known today. Our results on the geometry of the 27-dimensional module for Chevalley groups of type E 6 and of the 56-dimensional module for Chevalley groups of type E 7 pave the way to much more general results such as description of subgroups normalized by some of elementary matrices. Groups of types E 8 and F 4 can be handled in essentially the same style, and we intend to return to these cases in our subsequent publications. However, from a technical viewpoint the proofs in these cases are noticeably more involved because these groups do not have microweight representations. §1. IntroductionThe proofs of structure theorems are based either on reduction of the dimension of the ground ring, or on reduction of the rank of the group, or eventually on a combination of both. The early proofs for classical groups were based on reduction of rank [71], [17]-[19], and [23,24,20,3,9]; the dimension reduction proofs appeared later [60], [62]-[64], and [44, 45, 33]. At the same time for exceptional groups the first published proofs leant upon reduction of dimension (see [25,27,28,30,58,61]). The paper [30] by E. Abe and K. Suzuki was a near miss as far as the normal structure theorem for Chevalley groups is concerned (see also [25]). In fact, a somewhat weaker result was proved in [30], namely, a description of the normal subgroups of the elementary Chevalley group E(Φ, R) over a class of rings subject to a certain finiteness condition. However, as was

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A Study of Suslin Matrices: Their Properties and Uses

Rao

Jose

2016

Algebra and Its Applications

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K1 of Chevalley groups are nilpotent

Cited by 84 publications

References 22 publications

Normalizer of the Chevalley group of type ${\mathrm E}_6$

Normalizer of the Chevalley group of type ${\mathrm E}_6$

An $\mathrm {A}_2$-proof of structure theorems for Chevalley groups of types $\mathrm {E}_6$ and $\mathrm {E}_7$

A Study of Suslin Matrices: Their Properties and Uses

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