Another presentation for symplectic Steinberg groups

Lavrenov, Andrei

doi:10.1016/j.jpaa.2014.12.021

Cited by 20 publications

(32 citation statements)

References 40 publications

(54 reference statements)

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“…x −2ε i ) is the generator corresponding to the long root 2ε i (resp. to −2ε i ); this generator is independent of i and central (see Lemma 2.1 below; see also [14]).…”

Section: The Steinberg Group St(c N Z)mentioning

confidence: 97%

Braid groups and symplectic Steinberg groups

Digne¹,

Kassel²

2022

Preprint

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“…x −2ε i ) is the generator corresponding to the long root 2ε i (resp. to −2ε i ); this generator is independent of i and central (see Lemma 2.1 below; see also [14]).…”

Section: The Steinberg Group St(c N Z)mentioning

confidence: 97%

Braid groups and symplectic Steinberg groups

Digne¹,

Kassel²

2022

Preprint

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“…To define the elementary symplectic group Ep 2n (R) instead of T ij (a) one can use "basis-independent" ESD-transformations T (u, v, a) defined by w → w + u( v, w + a u, w ) + v u, w as a set of generators. Here u, v ∈ R 2n , u, v = 0, a ∈ R. See section 1 of [6] for details. On the level of Steinberg groups, this idea leads to the following presentation, inspired by van der Kallen's paper [3].…”

Section: Absolute Steinberg Groupsmentioning

confidence: 99%

“…The paper is organised as follows. In the first section we recall results of [6], where a "basis-free" presentation of the symplectic Steinberg group id given. We make an essential use of these results in the Section 4.…”

Section: Introductionmentioning

confidence: 99%

A local-global principle for symplectic $\mathrm K_2$

Lavrenov¹

2016

Preprint

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“…The first ingredient is the so-called centrality of the non-stable K 2 -functor associated to G over a local ring. For Chevalley-Demazure groups over local rings, the corresponding result is due to M. R. Stein [Stein,Theorem 2.13]; for groups of rank ≥ 3 of simply laced or symplectic type it is even known over general commutative rings [vK77,La15,LaSi17]. V. Deodhar defined Steinberg groups for isotropic reductive groups over fields in terms of a minimal parabolic subgroup of G, and established the centrality of the corresponding non-stable K 2 -functors [Deo,Prop.…”

Section: Introductionmentioning

confidence: 99%

On the congruence kernel of isotropic groups over rings

Stavrova

2020

Trans. Amer. Math. Soc.

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Let R be a connected noetherian commutative ring, and let G be a simply connected reductive group over R of isotropic rank ≥ 2. The elementary subgroup E(R) of G(R) is the subgroup generated by U P + (R) and U P − (R), where U P ± are the unipotent radicals of two opposite parabolic subgroups PWe prove that the congruence kernel of E(R), defined as the kernel of the natural homomorphism E(R) → E(R) between the profinite completion of E(R) and the congruence completion of E(R) with respect to congruence subgroups of finite index, is central in E(R). In the course of the proof, we construct Steinberg groups associated to isotropic reducive groups and show that they are central extensions of E(R) if R is a local ring. , and the research program 6.38.74.2011 "Structure theory and geometry of algebraic groups and their applications in representation theory and algebraic K-theory" at St. Petersburg State University. Proof. See [St14, Theorem 2.1]. Definition 2.5. Under the assumptions of Theorem 2.4, we call the subgroup E(A) = E P (A), where P is a strictly proper parabolic subgroup of G, the elementary subgroup of G(A).2.2. Abstract relative roots. In order to better understand systems of relative roots Φ(S, G) of Definition 2.2, we consider the abstract systems of relative roots derived from abstract root systems in the sense of [Bou]. The corresponding notion was first introduced in [PeSt09].Let Φ be a reduced root system in the sense of [Bou] with an inner product (−, −). Let Π = {a 1 , . . . , a l } be a fixed system of simple roots of Φ; if Φ is irreducible, we assume that the numbering follows Bourbaki [Bou]. Let D be the Dynkin diagram of Φ. We identify nodes of D with the corresponding simple roots in Π.Definition 2.6. Let Γ ≤ Aut (D) be a subgroup, and let J ⊆ Π be a Γ-invariant subset. Consider the projectionProof. The proof of (i) and (ii) for the case of maximal roots is literally the same as in [KaSt, Lemma 1], which deals with the case where α is a simple relative root. The case of minimal roots is symmetric. To establish (iii), we observe that Ψ = Φ Π,Γ is naturally a root system, and Ψ π Π,Γ (J),{id} = Φ J,Γ . Applying (i) to Ψ π Π,Γ (J),{id} , we conclude that the images under π Π,Γ of all Π-maximal (respectively, Π-minimal) roots in π −1 J,Γ (α) coincide. By [PeSt09, Lemma 3] this implies that Γ acts transitively on such roots. The claim (iv) follows similarly from (ii).Lemma 2.8. Assume that rank(Φ J,Γ ) ≥ 2 and it is irreducible. Let α, β ∈ Φ J,Γ be two relative roots. Let a max ∈ π −1 J,Γ (α) be a Π-maximal root, and let b min ∈ π −1 J,Γ (β) be a Π-minimal root. Let a 1 , . . . , a n ∈ Φ + be a sequence of roots such that b min = a max − a 1 − . . . − a n ,

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Another presentation for symplectic Steinberg groups

Cited by 20 publications

References 40 publications

Braid groups and symplectic Steinberg groups

Braid groups and symplectic Steinberg groups

A local-global principle for symplectic $\mathrm K_2$

On the congruence kernel of isotropic groups over rings

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