Characterizations of strongly simply connected polynomial growth algebras
type: Item
instance of: scholarly article
title: Characterizations of strongly simply connected polynomial growth algebras
zbMATH Open document ID: 0872.16010
DOI: 10.1007/BF01198069
published in: Archiv der Mathematik
publication date: 1996-11-22T00:00:00Z
review text: Let \(K\) be an algebraically closed field and let \(A\) be a connected basic finite dimensional \(K\)-algebra viewed as a quotient \(KQ/I\) of a path algebra \(KQ\) of a quiver \(Q=(Q_0,Q_1)\) (i.e. a finite oriented graph) modulo an admissible ideal \(I\). Here \(Q_0\) is the set of vertices of \(Q\) and \(Q_1\) is the set of arrows of \(Q\). The algebra \(A\) is said to be strongly simply connected if \(Q\) has no oriented cycles and the first Hochschild cohomology space \(H^1(C,C)\) vanishes for every full convex subcategory \(C\) of \(A\). The algebra \(A\) is said to be tame of polynomial growth if there exists an integer \(m\) such that the indecomposable \(A\)-modules occur, in each dimension \(d\), in a finite number of discrete and at most \((d+1)^m\) one-parameter families. The main aim of the paper is to give a characterization of strongly simply connected tame algebras of polynomial growth in terms of an integral Tits quadratic form \(q_A:\mathbb{Z}^{Q_0}\to\mathbb{Z}\) associated with \(A\). Given a finite dimensional right \(A\)-module \(X\) we denote by \(\mathbf{dim} X\in\mathbb{Z}^{Q_0}\) the dimension vector of \(X\). By a support algebra of \(X\) we shall mean the full subcategory \(B(X)\) of \(A\) formed by the vertices of the support \(\text{supp}(X)=\{i\in Q_0;\;X(i)\neq 0\}\). We shall also use the number \(m(X)=\min\{\dim_KX(i);\;i\in\text{supp}(X)\}\). The main results of the paper assert that if \(A\) is a finite dimensional strongly simply connected algebra then the following conditions are equivalent: (1) \(A\) is tame of polynomial growth, (b) the integral Tits quadratic form \(q_A\) is weakly non-negative (that is, \(q_A(v)\geq 0\) for all \(v\in\mathbb{N}^{Q_0}\)) and \(m(X)=1\) for every indecomposable \(A\)-module \(X\) with \(q_A(\mathbf{dim} X)\geq 2\), (3) \(q_A(\mathbf{dim} X)\in\{0,1\}\) for every indecomposable \(A\)-module \(X\) with \(m(X)\geq 2\), (4) for every indecomposable \(A\)-module \(X\) with \(m(X)\geq 3\) and \(q_A…
reviewed by: Daniel Simson
Mathematics Subject Classification ID: 16G60
Mathematics Subject Classification ID: 15A63
Mathematics Subject Classification ID: 16G20
zbMATH DE Number: 937027
zbMATH Keywords: quivers with relations
zbMATH Keywords: path algebras
zbMATH Keywords: connected basic finite dimensional algebras
zbMATH Keywords: simply connected tame algebras of polynomial growth
zbMATH Keywords: Tits quadratic forms
zbMATH Keywords: Euler forms
zbMATH Keywords: indecomposable modules
zbMATH Keywords: tilted algebras of Euclidean type
author: José A. de la Peña
author: Andrzej Skowroński
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01198069
OpenAlex ID: W2001340564
description: scientific article; zbMATH DE number 937027