The Tits and Euler forms of a tame algebra
type: Item
instance of: scholarly article
title: The Tits and Euler forms of a tame algebra
zbMATH Open document ID: 0941.16010
author: Andrzej Skowroński
author: José A. de la Peña
published in: Mathematische Annalen
publication date: 2000-05-18T00:00:00Z
review text: The use of quadratic forms in representation theory of algebras has lead to important results in the area, in particular in the study of the representation type of finite dimensional algebras. In the paper under review, the authors study some properties of the Euler and the Tits forms associated to a tame (strongly) simply connected algebra. Recall that the Tits form of an algebra gives informations related to geometrical aspects while the Euler form reflects the homological behavior of the module category. Let \(k\) be an algebraically closed field and \(A=kQ/I\) be a strongly simply connected algebra, where \(Q\) a finite quiver and \(I\) is an admissible ideal of the path algebra \(kQ\). Denote by \(q_A\) and by \(\chi_A\) the Tits and the Euler forms of \(A\), respectively. The authors show that if \(A\) is of polynomial growth and \(X\) is an indecomposable \(A\)-module, then: (i) \(0\leq q_A(\mathbf{dim} X)\leq 18n\), where \(n\) indicates the number of vertices of \(Q\); (ii) \(0\leq\chi_A(\mathbf{dim} X)\leq 2+e\), where \(e\) indicates the number of projective-injective indecomposable modules; and (iii) if \(X\) is faithful, then \(q_A(\mathbf{dim} X),\chi_A(\mathbf{dim} X)\leq 2\).
Mathematics Subject Classification ID: 16G60
Mathematics Subject Classification ID: 15A63
Mathematics Subject Classification ID: 16G20
zbMATH DE Number: 1347388
zbMATH Keywords: Tits forms
zbMATH Keywords: Euler forms
zbMATH Keywords: tame algebras
zbMATH Keywords: finite dimensional algebras
zbMATH Keywords: strongly simply connected algebras
zbMATH Keywords: quadratic forms
zbMATH Keywords: path algebras
zbMATH Keywords: algebras of polynomial growth
zbMATH Keywords: indecomposable modules
reviewed by: Flávio U. Coelho
MaRDI profile type: Publication
full work available at URL: s002080050317
OpenAlex ID: W1975229172
DOI: 10.1007/S002080050317
description: scientific article; zbMATH DE number 1347388