Mal'cev-Neumann rings and noncrossed product division algebras.
type: Item
instance of: scholarly article
zbMATH Open document ID: 1263.16022
DOI: 10.1142/S0219498811005804
author: Cécile Coyette
publication date: 2012-06-25T00:00:00Z
full work available at URL: https://arxiv.org/abs/1103.0470
Mathematics Subject Classification ID: 16K20
Mathematics Subject Classification ID: 16S35
Mathematics Subject Classification ID: 16W60
zbMATH DE Number: 6049618
zbMATH Keywords: Malcev-Neumann series
zbMATH Keywords: division algebras
zbMATH Keywords: noncrossed products
zbMATH Keywords: crossed products
zbMATH Keywords: local invariants
zbMATH Keywords: generalized crossed products
zbMATH Keywords: central simple algebras
MaRDI profile type: Publication
arXiv ID: 1103.0470
cites work: On central division algebras
cites work: Noncrossed Products and Nonabelian Crossed Products Over ℚ(T) and ℚ((T))
cites work: Noncrossed products over \(k_{\mathfrak p}(t)\)
cites work: A twisted Laurent series ring that is a noncrossed product.
cites work: Division algebras over Henselian fields.
OpenAlex ID: W2964093415
published in: Journal of Algebra and its Applications
title: Mal'cev-Neumann rings and noncrossed product division algebras.
review text: A finite dimensional division algebra is a crossed product if it contains a maximal subfield over the center. In 1972 Amitsur showed that for degree divisible by an odd square or by \(8\), the generic division algebra is a noncrossed product. As part of the theory of central simple algebras over special fields, there is interest in constructing more explicit noncrossed products. Several examples were given, most notably by Brussel and Hanke. In this paper, Coyette shows that Malcev-Neumann algebras associated to generalized (Abelian) crossed products are another interesting source of noncrossed products.NEWLINENEWLINE Let \(A\) be a central simple algebra over a field \(F\), and assume \(A\) contains a (nonmaximal) field \(F'\), Galois over \(F\), with an Abelian Galois group \(G=\text{Gal}(F'/F)\). Let \(A'=C_A(F')\) denote the centralizer of \(F'\) in \(A\). Fix, for each \(\sigma\in G\), an element \(u_\sigma\in A\) such that \(u_\sigma^{-1}xu_\sigma=\sigma(x)\) for \(x\in F'\). Let \(\Gamma\) be a totally ordered Abelian group with a surjection \(\varepsilon\colon\Gamma\to G\). The Malcev-Neumann algebra \(\mathcal D=A'((\Gamma))\) is the ring of formal series \(\sum_{\gamma\in\Gamma}a_\gamma r_\gamma\) in the variables \(a_\gamma\) with coefficients \(r_\gamma\in A'\) and with well ordered support, subject to the relations \(r(a_\gamma u_{\varepsilon(\gamma)}^{-1})=(a_\gamma u_{\varepsilon(\gamma)}^{-1})r\) for \(r\in A'\) and \(a_\gamma a_\delta=a_{\gamma+\delta}u^{-1}_{\varepsilon(\gamma+\delta)}u_{\varepsilon(\gamma)}u_{\varepsilon(\delta)}\). For \(\Gamma=\mathbb Z\) one obtains a twisted Laurent series ring in one variable over \(A'\).NEWLINENEWLINE Assume \(A'\) is a division algebra. Then \(\mathcal D\) is a valued division algebra with value group \(\Gamma\) and residue ring \(A'\), whose center is \(F((\Gamma_0))\) for \(\Gamma_0=\ker(\varepsilon)\), and with degree equal to the degree of \(A\). The key result is that Galois maximal subfields of \(\mat…
reviewed by: Uzi Vishne
description: scientific article; zbMATH DE number 6049618