Decomposition of involutions on inertially split division algebras
type: Item
instance of: scholarly article
title: Decomposition of involutions on inertially split division algebras
zbMATH Open document ID: 0985.16014
author: B. A. Sethuraman
author: Patrick Morandi
published in: Mathematische Zeitschrift
publication date: 2002-02-12T00:00:00Z
full work available at URL: http://hdl.handle.net/10211.3/171010
review text: Let \(F\) be a Henselian valued field with residue field \(\overline F\) of characteristic not~\(2\) and let \(S\) be an \(F\)-central (finite-dimensional) division algebra which is split by an inertial extension of \(F\). Assume \(\sigma\) is an involution on \(S\) (i.e., an anti-automorphism of period~\(2\)) which is the identity on an inertial lift in \(S\) of the center of the residue algebra \(\overline S\). The authors prove necessary and sufficient conditions for \(S\) to contain a \(\sigma\)-stable quaternion \(F\)-subalgebra or to decompose into a tensor product of \(\sigma\)-stable quaternion algebras, in terms of the decomposability of an associated central simple algebra over the residue field \(\overline F\). Their results extend to the general case of inertially split algebras over a Henselian field work of \textit{H.~Dherte} [Math. Z. 216, No. 4, 629-644 (1994; Zbl 0812.16023)] on Malcev-Neumann series division algebras.
Mathematics Subject Classification ID: 16K20
Mathematics Subject Classification ID: 16W60
Mathematics Subject Classification ID: 16W10
zbMATH DE Number: 1533440
zbMATH Keywords: valuations on division algebras
zbMATH Keywords: involutions
zbMATH Keywords: central finite-dimensional division algebras
zbMATH Keywords: tensor products of quaternion algebras
zbMATH Keywords: central simple algebras
zbMATH Keywords: Malcev-Neumann series division algebras
reviewed by: Jean-Pierre Tignol
MaRDI profile type: Publication
OpenAlex ID: W2015824562
DOI: 10.1007/S002090000131
Recommended article: Decomposition of Mal'cev-Neumann division algebras with involution
Recommended article: Q4861157
Recommended article: Indecomposable Division Algebras
Recommended article: A certain decomposition of infinite invertible matrices over division algebras
Recommended article: Decomposability for division algebras of exponent two and associated forms
Recommended article: On the structure of split involutive Lie algebras
Recommended article: Q5370648
Recommended article: Decomposition of algebras with involution in characteristic 2
Recommended article: Invariant splitting in Jordan and alternative algebras
Recommended article: Division algebras and maximal orders for given invariants
description: scientific article; zbMATH DE number 1533440