Relation modules of prounipotent groups
type: Item
instance of: scholarly article
title: Relation modules of prounipotent groups
zbMATH Open document ID: 0633.20029
DOI: 10.1016/0021-8693(87)90163-3
published in: Journal of Algebra
publication date: 1987-01-01T00:00:00Z
review text: The object is to develop for prounipotent groups a concept analogous to that one of relation modules from discrete group theory. Let G be a prounipotent group over the algebraically closed field k of characteristic zero. G has a presentation as quotient of a free prounipotent group F by a normal prounipotent subgroup R. Denote by \(\bar R\) the quotient of R by the Zariski-closure (R,R) of its derived group. G acts induced by conjugation on \(\bar R.\) The first part of the paper consists of the construction of a suitable resolution of k, in which \(\bar R\) occurs as the second syzygy. Assume that F is free on \(d<\infty\) generators, however allow G to be infinitely related. Then the resolution has the form \[ k\ll G\gg^ r\to k\ll G\gg^ d\to k\ll G\gg \to k\to 0. \] The terms of the resolution are direct products of the complete group algebra (the dual of the coordinate ring of G) \(k\ll G\gg\), the cardinal r can be infinite. So \(k\ll G\gg\) is the analogue to the group ring in the discrete case. In the second part the author analyzes the structure of \(\bar R\) as \(k\ll G\gg\)-module and the structure of the group F/(R,R). In particular the Magnus embedding of F/(R,R) is constructed. Finally it is shown that higher relation modules (the other quotients of the closed lower central series of R) are also submodules of direct products of copies of \(k\ll G\gg.\) All constructions depend on the properties of prounipotent groups developed by \textit{A. Lubotzky} and the author [J. Algebra 74, 76-95 (1982; Zbl 0473.20031), ibid. 80, 323-349 (1983; Zbl 0503.20008), Am. J. Math. 107, 531-553 (1985; Zbl 0565.20023), J. Algebra 95, 46-58 (1985; Zbl 0596.20033)]. In this sense the article may be regarded as a continuation of this work.
Mathematics Subject Classification ID: 20G10
Mathematics Subject Classification ID: 14L15
Mathematics Subject Classification ID: 20E18
Mathematics Subject Classification ID: 20J05
Mathematics Subject Classification ID: 20F14
Mathematics Subject Classification ID: 20C07
zbMATH DE Number: 4029830
zbMATH Keywords: prounipotent groups
zbMATH Keywords: relation modules
zbMATH Keywords: presentation
zbMATH Keywords: free prounipotent group
zbMATH Keywords: syzygy
zbMATH Keywords: resolution
zbMATH Keywords: direct products
zbMATH Keywords: complete group algebra
zbMATH Keywords: coordinate ring
zbMATH Keywords: Magnus embedding
zbMATH Keywords: closed lower central series
author: Andy R. Magid
MaRDI profile type: Publication
full work available at URL: 0021-8693(87)90163-3
OpenAlex ID: W1971885578
cites work: Algebraic groups and Hopf algebras
cites work: Q5588717
cites work: Cohomology of unipotent and prounipotent groups
cites work: Free prounipotent groups
cites work: Cohomology, Poincare Series, and Group Algebras of Unipotent Groups
cites work: Q4145882
cites work: Coordinate rings of prounipotent groups
cites work: Annihilators of relation modules. II
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description: scientific article; zbMATH DE number 4029830