Metaplectic extensions of quaternion skew fields
type: Item
instance of: scholarly article
title: Metaplectic extensions of quaternion skew fields
zbMATH Open document ID: 0607.12006
DOI: 10.1007/BF01160480
author: Joachim Klose
published in: Mathematische Zeitschrift
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173810
review text: The author considers central extensions of \(D_{\mathbb A}^{\times}\) where \(D\) is a quaternion algebra over a number field \(k\). The extension is assumed to be a ``product'' over the places of \(k\) and to split over \(D^{\times}\). In a previous paper [ibid. 191, 261--282 (1986; Zbl 0563.12014)] the author has studied the corresponding local problem. There are two obvious constructions of cocycles with values in \(\mu_ n(k)\)- one by restriction of the Kubota cocycle from \(\mathrm{GL}_ 2(k_{\mathbb A}')\) where \(k'\) is a splitting field and the other by lifting back a cocycle of \(\mathrm{GL}_ 1(k_{\mathbb A})\) by the reduced norm. The author shows that all central topological extensions by \(\mu_ n(k)\) are generated by these when \(n\) is odd. The proof is based on his previous characterization of the local extensions by commutators and a version of Moore's theorem. Also included are some remarks on the case n being a power of 2, which is more complicated.
Mathematics Subject Classification ID: 11R52
Mathematics Subject Classification ID: 20G30
Mathematics Subject Classification ID: 16Kxx
Mathematics Subject Classification ID: 20G10
zbMATH DE Number: 3981289
zbMATH Keywords: central extensions
zbMATH Keywords: quaternion algebra
zbMATH Keywords: constructions of cocycles
reviewed by: Samuel James Patterson
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3981289