Grids and the arithmetics of Jordan pairs
type: Item
instance of: scholarly article
title: Grids and the arithmetics of Jordan pairs
zbMATH Open document ID: 0945.17007
author: Holger P. Petersson
published in: Journal of Algebra
publication date: 2000-10-05T00:00:00Z
review text: The author uses \textit{E. Neher}'s theory of grids [Jordan triple systems by the grid approach, Lect. Notes Math. 1280, Springer (1987; Zbl 0621.17001)]\ to derive several new results on orders in finite-dimensional Jordan pairs over local fields. In the first part, the concept of a norm on a Jordan pair \(V\) over a valued field \((K,\Gamma,\lambda)\) is introduced, as follows. Let \(\Delta\) be a totally ordered abelian group containing \(\Gamma\) as a totally ordered subgroup. A \(\lambda\)-norm on \(V\) with values in \(\Delta\) is a pair \(\rho=(\rho^+,\rho^-)\) of mappings \(\rho^\varepsilon:V^\varepsilon\to\Delta_\infty\) (\(\varepsilon=\pm\)) satisfying the following conditions: (1) \(\rho^\varepsilon(x)=\infty \Leftrightarrow x=0\), (2) \(\rho^\varepsilon(x+x')\geq\min(\rho^\varepsilon(x),\rho^\varepsilon(x'))\), (3) \(\rho^\varepsilon(Q(x)y)\geq 2\rho^\varepsilon(x)+\rho^{-\varepsilon}(y)\), and (4) \(\rho^\varepsilon(ax)=\lambda(a)+\rho^\varepsilon(x)\). The following ``Norm Theorem'' guarantees the existence of norms in a particular case: Let \(V\) be a Jordan pair over a Henselian field \((K,\Gamma,\lambda)\). Let \(G\) be a covering division grid in \(V\) whose Peirce components are \(2\)-finite-dimensional (i.e.\ every subpair generated by two elements \(x,y\) satisfying \(x^+=y^+\) or \(x^-=y^-\) is finite-dimensional). Then \(G\) canonically determines a \(\lambda\)-norm \(\rho\) of \(V\) with values in \(\Gamma\otimes_{\mathbb Z}\mathbb Q\). The main idea is to define \(\rho\) on the Peirce components of \(V\) relative to \(G\); since these Peirce components are Jordan division pairs, Niggemann's results on valuations of Jordan division rings [\textit{D. Niggemann}, Allgemeine Bewertungstheorie für Jordan-Divisionsringe, Thesis, Hagen (1995)] can be used for this task. The Norm Theorem is used to classify saturated maximal orders (i.e.\ maximal orders containing a covering division grid): Let \(V\) be a finite-dimensional Jordan pair over the loca…
Mathematics Subject Classification ID: 17C10
Mathematics Subject Classification ID: 17C55
Mathematics Subject Classification ID: 17A80
zbMATH DE Number: 1270875
zbMATH Keywords: Jordan pairs over local fields
zbMATH Keywords: valuations of Jordan division pairs
zbMATH Keywords: norms of Jordan pairs
zbMATH Keywords: grids
zbMATH Keywords: saturated order
zbMATH Keywords: weakly separable order
zbMATH Keywords: anisotropic part
zbMATH Keywords: weakly unramified Jordan pair
reviewed by: Harald Löwe
MaRDI profile type: Publication
full work available at URL: jabr.1998.7663
OpenAlex ID: W2006779329
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DOI: 10.1006/JABR.1998.7663
description: scientific article; zbMATH DE number 1270875