On centrality ofK2for Chevalley groups of typeEℓ

Abstract

For a root system Φ of type E ℓ and arbitrary commutative ring R we show that the group K 2 (Φ, R) is contained in the center of the Steinberg group St(Φ, R). In course of the proof we also demonstrate an analogue of Quillen-Suslin local-global principle for K 2 (Φ, R).It is classically known that K 2 (Φ, R) is contained in the center of St(Φ, R) if R is a local ring and Φ has rank ≥ 2 (see [14, Theorem 2.13]). One of the standard conjectures in the theory of Chevalley groups over rings asserts that St(Φ, R) is a central extension of E(Φ, R) for any commutative ring R provided that the rank of Φ is sufficiently large. We refer to this conjecture as centrality of K 2 . It can be regarded as a "K 2 -analogue" of Taddei's normality theorem. For Chevalley groups of rank 2 centrality of K 2 fails already for 1-dimensional R (see [25, Theorem 1]).W. van der Kallen in [5] and recently A. Lavrenov in [7] proved centrality of K 2 for Φ = A ℓ , C ℓ , ℓ ≥ 3. The main technical ingredient of both proofs is the so-called method of "another presentation" which consists in presenting St(Φ, R) as a group with a set of generators modeling elements of root type (see [22, § 4] for the idea of the definition). For example, in the linear case the elements of root type are exactly the usual linear transvections, see Section 2.3. The key advantage of this method lies in the fact that it allows one to define an action of G(Φ, R) on St(Φ, R) which turns the canonical map φ into a crossed module.Recall that a crossed module is a triple C = (N, M, µ) consisting of an abstract group N acting on itself by conjugation, an N-group M and a group morphism µ : M → N which preserves the action of N and satisfies Peiffer identity µ(m) · m ′ = mm ′ m −1 , m, m ′ ∈ M. Date: October 19, 2018.

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

“…The correct approach to relative Steinberg groups is described in [4,7,8]. But for splitting ideals we can define it in the following naive way.…”

“…For the next lemma the proof of Lemma 16 of [8] works verbatim. There are two references in that proof: instead of Lemma 8 of [8] use Lemma 2.2, and instead of Lemma 15 of [8] use Lemma 3.1.…”

“…Presently, K 1 -analogue is known in much larger generality, namely, for isotropic reductive groups [9] and in the framework of Stepanov's universal localisation [10]. As opposed to that, K 2 -analogues are proven only for GL n [14] and, recently, for Chevalley groups of type E l [8].…”

Explicit presentation of relative Steinberg groups

“…There is another definition of St(n, R, I) by M. Tulenbaev [6] if R is commutative, but in terms of generators X vw parametrized by vectors v, w satisfying some conditions instead of the elementary generators z ij (a, p). By [5], both definitions give the same groups for commutative rings.…”

“…For relative simply laced Steinberg groups St(Φ; R, I) we use a definition from [5]. It is well-known that relative Steinberg groups (both linear and simply laced) are generated by the elementary conjugates z ij (a, p) = xji (p) x ij (a) as abstract groups (see, for example, [5, lemma 5] for the case of simply-laced groups).…”