Steinberg groups for Jordan pairs
type: Item
instance of: scholarly article
title: Steinberg groups for Jordan pairs
zbMATH Open document ID: 1215.17019
author: Ottmar Loos
author: Erhard Neher
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2010-09-06T00:00:00Z
review text: Given a locally finite root system \(\phi\), for any two elements \(\alpha\) and \(\beta\) in \(\phi\), the authors define \(((\alpha,\beta)):=\phi\cap(\mathbb N_+\alpha+\mathbb N_+\beta)\) where \(\mathbb N_+:=\{x\in\mathbb N: x>0\}\). A pair \((\alpha, \beta)\) of roots is said to be nilpotent if \(p\alpha+q\beta\neq 0\) for all \(p,q\in\mathbb N_+\). A \textsl{group with commutator relations} of type \(\phi\) is a group \(G\) provided with a family of subgroups \((U_\alpha)_{\alpha\in\phi}\) (called the \textsl{root groups}) which generate \(G\) and for which the commutator relations \(((U_\alpha,U_\beta))\subset U_{((\alpha,\beta))}\) hold for all nilpotent pairs \((\alpha,\beta)\). The notations \(U_{((\alpha,\beta))}\) stands for the subgroup of \(G\) generated by all \(U_\gamma\) where \(\gamma\in((\alpha,\beta))\) (plus some additional condition if \(\phi\) is not reduced). Given any two groups with \(\phi\)-commutator relations, the notion of a morphism \(\phi: (G,U_\alpha)\to(\tilde G, {\tilde U}_\alpha)\) is the obvious one: root groups are \(\phi\)-invariant. In the paper, the authors use the notion of \textsl{strong morphism} which roughly speaking amounts to requiring that each restriction \(\phi: U_\alpha\to {\tilde U}_\alpha\) is bijective and that \(G\) has the same generators and commutator relations than \(\tilde G\). For a fixed group with \(\phi\)-commutator relations \((\tilde G, {\tilde U}_\alpha)\) we can consider the category whose objects are the pairs \((G,s)\) where \(G=(G,U_\alpha)\) is a group with \(\phi\)-commutator relations and \(s: G\to \tilde G\) is a strong morphism. An initial element for this category is called a \textsl{Steinberg group} of \(\tilde G\). Consider a Jordan pair \(V\) over a commutative (associative) unitary ring \(k\) and its Tits-Kantor-Koecher algebra \(g(V)=g=g_{-1}\oplus g_0\oplus g_1\) (where \(g_{\pm 1}=V^{\pm}\) as \(k\)-modules.) Since \(\text{ad}(x)^3=0\) for \(x\in g_{\pm}\) we can exponentiate \(\text…
Mathematics Subject Classification ID: 17C30
Mathematics Subject Classification ID: 17C27
zbMATH DE Number: 5779540
zbMATH Keywords: Steinberg groups
zbMATH Keywords: Jordan pairs
zbMATH Keywords: locally finite root systems
zbMATH Keywords: central extensions
reviewed by: Candido Martín González
MaRDI profile type: Publication
full work available at URL: j.crma.2010.07.012
OpenAlex ID: W4299562523
cites work: Q3941610
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DOI: 10.1016/J.CRMA.2010.07.012
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description: scientific article; zbMATH DE number 5779540