Noetherian equations for matrices
type: Item
instance of: scholarly article
title: Noetherian equations for matrices
zbMATH Open document ID: 0912.12005
DOI: 10.1023/A:1005960503519
author: Gerhard Pazderski
published in: Acta Applicandae Mathematicae
publication date: 1999-03-22T00:00:00Z
review text: Let \(L\) be a finite Galois extension of the field \(K\) and denote by \(G\) the Galois group \(\text{Gal}(L/K)\). The author investigates two kinds of ``Noetherian equations''. The first kind is the set of solutions of the matrix equation \[ X_{\sigma \tau}=X_\sigma^\tau X_\tau, \] where \(\sigma,\tau\) are in \(G\), \(X_\sigma\) is in \(GL_n(L)\) for a given natural number \(n\), and \(X_\sigma^\tau \) is obtained from \(X_\sigma\) by applying \(\tau\) to all entries of \(X_\sigma\). The map which associates \(\sigma\) to \(X_\sigma\) is called by the author a crossed representation of \(G\) over \(L\). The second kind is the set of solutions of \[ X_{\sigma \tau} f_{\sigma, \tau}=X_\sigma^\tau X_\tau, \] where \(X_\sigma\) is in \(GL_n(L)\) and \(f_{\sigma, \tau}\) is in \(L^\times\) (the invertible elements of \(L\)). In fact it is the previous equation with the solutions in \(PGL_n(L)\), and the author calls the map which associates \(\sigma\) to \(X_\sigma\), crossed-projective representation of \(G\) over \(L\). We can see that \(f_{\sigma,\tau}\) is in fact a 2-cocycle over \(G\) with coefficient in \(L^\times\). With a 2-cocycle \(f_{\sigma,\tau}\), we can form the crossed-product of \(L\) and \(G\) over \(K\) relative to \(f\), denoted by \((L,G,f)\), which is a central simple algebra over \(K\). Then the author shows (Corollary 1, p. 318) that all the solutions of the first equation are given by \(X_\sigma=Q^{-\sigma} Q\), where \(Q\) ranges over the elements of \(GL_n(L)\), and he shows (Proposition 4, p. 320) that there is a correspondence, up to equivalence, between 2-cocycles and solutions of the second equation. He also shows (Proposition 5, p. 320) that the minimal degree of a solution of the second equation (i.e. the smallest \(n\) such that \(X_\sigma\) is in \(GL_n(L)\)) coincides with the Schur index of the algebra \((L,G,f)\). The author also gives some properties in the case of cyclic extensions. Rephrased in the cohomological language, the m…
reviewed by: Philippe Elbaz-Vincent
Mathematics Subject Classification ID: 12G05
Mathematics Subject Classification ID: 11E72
Mathematics Subject Classification ID: 16S35
Mathematics Subject Classification ID: 16K20
Mathematics Subject Classification ID: 15A24
zbMATH DE Number: 1218155
zbMATH Keywords: Noetherian equation
zbMATH Keywords: matrix equations
zbMATH Keywords: central simple algebra
zbMATH Keywords: crossed representation
zbMATH Keywords: crossed projective representation
zbMATH Keywords: cyclic algebra
zbMATH Keywords: Schur index
zbMATH Keywords: crossed product
zbMATH Keywords: Galois cohomology
MaRDI profile type: Publication
full work available at URL: a:1005960503519
OpenAlex ID: W32112657
description: scientific article; zbMATH DE number 1218155