Pseudo representations
type: Item
instance of: scholarly article
title: Pseudo representations
zbMATH Open document ID: 0863.16012
DOI: 10.1007/BF01445251
author: Louise Nyssen
published in: Mathematische Annalen
publication date: 1997-04-23T00:00:00Z
full work available at URL: https://eudml.org/doc/165454
review text: Let \(A\) be a ring and \(R\) an \(A\)-algebra. A representation of \(R\) over \(A\) of dimension \(d\) is a homomorphism of \(A\)-algebras \(\rho:R\to M_d(A)\). A pseudo-representation of \(R\) over \(A\) of dimension \(d\) is a central \(A\)-linear form \(T:R\to A\), whose value at \(1_R\) is \(d\) and for which a certain fundamental identity is true. For example, the trace of a faithful representation is a pseudo-representation of the same dimension. The purpose of the article is to prove a converse proposition. Assume that \(A\) is a Henselian separable ring with residue field \(F\) and maximal ideal \(\mathfrak m\), and let \(T\) be a pseudo-representation of \(R\) over \(A\) of dimension \(d\). If there exists an absolutely irreducible representation \(\overline\rho:R\otimes_A F\to M_d(F)\) such that \(T\bmod{\mathfrak m}=\text{tr }\overline\rho\), then \(T\) is the trace of a faithful representation \(\rho:R\to M_d(A)\) which is unique up to conjugation and makes the diagram: \[ \begin{tikzcd} R \ar[r,"\rho"]\ar[d] & M_d(A)\ar[d]\\R\otimes_A F \ar[r,"\overline{\rho}"'] & M_d(F)\end{tikzcd} \] commute. Thanks to this, one can build explicitly the universal deformation ring of an absolutely irreducible representation over \(F\).
Mathematics Subject Classification ID: 16G30
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 16W20
zbMATH DE Number: 952631
zbMATH Keywords: homomorphisms of algebras
zbMATH Keywords: central linear forms
zbMATH Keywords: traces of faithful representations
zbMATH Keywords: pseudo-representations
zbMATH Keywords: absolutely irreducible representations
zbMATH Keywords: universal deformation rings
MaRDI profile type: Publication
cites work: Q3701517
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description: scientific article; zbMATH DE number 952631