Quaternion algebras with derivations
type: Item
instance of: scholarly article
title: Quaternion algebras with derivations
zbMATH Open document ID: 1485.12008
author: Amit Kulshrestha
author: Varadharaj Ravi Srinivasan
published in: Journal of Pure and Applied Algebra
publication date: 2021-09-02T00:00:00Z
review text: The goal of this paper is to study various questions regarding central simple algebras with derivations, with an emphasis on quaternion algebras. \textit{S. A. Amitsur} [Commun. Algebra 10, 797--803 (1982; Zbl 0486.16014)] proved that for every maximal subfield \(L\) of a central simple \(k\)-algebra \(A\), there is a derivation on \(A\) mapping \(L\) to \(L\). For example, when \(\operatorname{char}(k)=2\) and \(A\) is a quaternion algebra, the latter is generated over \(k\) by some \(x\) and \(y\) satisfying \(x^2+x=\alpha\), \(y^2=\beta\) and \(xy+yx=y\) where \(\alpha \in k\) and \(\beta\in k^\times\). In this case, the algebra \(A\) is endowed with a derivation \(\delta\) mapping any \(v\) to \(xv+vx\), and thus \(\delta\) stabilizes the quadratic subfield \(k(y)\) of \(A\). This raises the question of whether every derivation must stabilize a maximal subfield. This was answered in the negative in the paper under discussion (Proposition 4.6), where it is proved that every quaternion algebra over \(\mathbb{Q}\) admits a derivation stabilizing no quadratic subfield. Another theme in this paper is the study of splitting fields of central simple algebras with derivations, a notion introduced in [\textit{L. Juan} and \textit{A. R. Magid}, Proc. Am. Math. Soc. 136, No. 6, 1911--1918 (2008; Zbl 1223.12006)]. In Theorem 4.1 of the paper under discussion, the authors provide explicit constructions of differential fields that split differential quaternion algebras. They prove that there are (differential) fields of transcendence degree 1 over \(\mathbb{Q}\), splitting a differential quaternion \(\mathbb{Q}\)-algebra. They also prove that no algebraic extension splits a differential quaternion division \(\mathbb{Q}\)-algebra (Theorem 4.5). They conclude the paper by initiating a study of differential quaternion algebras over function fields. In Proposition 5.1, they show that those quaternion algebras over a function field \(k(t)\), which are not defined over \(k\), are o…
reviewed by: Adam Chapman
Mathematics Subject Classification ID: 12H05
Mathematics Subject Classification ID: 16H05
zbMATH DE Number: 7389870
zbMATH Keywords: differential fields
zbMATH Keywords: quaternion algebras
MaRDI profile type: Publication
full work available at URL: j.jpaa.2021.106805
OpenAlex ID: W3186153281
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DOI: 10.1016/J.JPAA.2021.106805
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description: scientific article; zbMATH DE number 7389870