Modules over crossed products of a division ring by a free Abelian group. I
type: Item
instance of: scholarly article
title: Modules over crossed products of a division ring by a free Abelian group. I
zbMATH Open document ID: 0958.16028
author: Christopher J. B. Brookes
author: J. R. J. Groves
published in: Journal of Algebra
publication date: 2001-01-24T00:00:00Z
review text: Let \(DA\) be a crossed product of \(A\), a free Abelian group of finite rank \(n\), over the division ring \(D\). This paper studies \(DA\)-modules \(M\) and uses the results to obtain interesting group theoretic applications. The main results are quite technical, so we will just state one. To start with, the dimension of \(M\), \(\dim M\), is defined to be the largest integer \(r\) such that \(M\) contains a nonzero free \(DB\)-submodule for some subgroup \(B\) of \(A\) with the rank of \(B\) equal to \(r\). This turns out to be the same as the Gelfand-Kirillov dimension of \(M\). Next, let \(R\) denote the additive group of real numbers and let \(\widehat A=\Hom(A,R)=R^n\) denote the set of characters \(\chi\colon A\to R\). If \(\chi\in\widehat A\), we say that \(M\) has a nontrivial \(\chi\)-filtration if there exists a family of \(D\)-spaces \(\{M_\mu\mid\mu\in R\}\) satisfying (1) \(M_\mu\supseteq M_\lambda\) if and only if \(\lambda\geq\mu\), (2) \(M=\bigcup_\mu M_\mu\) and \(M\neq M_\mu\), and (3) \(M_\mu\overline a=M_{\mu+\chi(a)}\) for all \(a\in A\). Then \(\Delta(M)\) is defined to be the subset of \(\widehat A\) consisting of \(0\) and those characters \(\chi\) such that \(M\) has a nontrivial \(\chi\)-filtration. With this notation, Theorem 4.4 asserts that \(\Delta(M)\subseteq\widehat A=R^n\) contains a rational convex polyhedron of dimension \(r=\dim M\). Note that a convex polyhedron is a finite intersection of closed half-spaces, and it is rational if all the defining half spaces have boundaries that are rational subspaces.
reviewed by: D. S. Passman
Mathematics Subject Classification ID: 16S35
Mathematics Subject Classification ID: 16D80
Mathematics Subject Classification ID: 20C07
Mathematics Subject Classification ID: 20F16
zbMATH DE Number: 1496002
zbMATH Keywords: crossed products
zbMATH Keywords: division rings
zbMATH Keywords: free Abelian groups
zbMATH Keywords: polyhedra
zbMATH Keywords: Gelfand-Kirillov dimension
MaRDI profile type: Publication
full work available at URL: jabr.1999.8194
OpenAlex ID: W4210735570
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DOI: 10.1006/JABR.1999.8194
description: scientific article; zbMATH DE number 1496002