Tame algebras and integral quadratic forms
type: Item
instance of: scholarly article
title: Tame algebras and integral quadratic forms
zbMATH Open document ID: 0546.16013
author: Claus Michael Ringel
published in: Lecture Notes in Mathematics
publication date: 1984-01-01T00:00:00Z
review text: This is an elegant introduction to some important parts of the new representation theory of finite dimensional algebras developed during the last 15 years. The author presents several methods of the modern representation theory in a unified and intelligible way. Apart from new versions of known results these notes contain the author's recent original results concerning interesting classes of tame algebras of finite global dimension. The main results contained in these notes are presented in terms of the following connection between indecomposable modules and quadratic forms. Let \(A\) be a basic finite dimensional K-algebra of finite global dimension over an algebraically closed field \(K\) having \(n\) nonisomorphic simple left \(A\)-modules. For each finite dimensional left \(A\)-module \(M\), the dimension vector \(\dim M\) of \(M\) is the \(n\)-tuple of multiplicities of fixed nonisomorphic simple \(A\)-modules appearing in the composition series of \(M\). Then \(\dim M\) is an element of the Grothendieck group \(K_0(A)\), identified with \(Z^n\), and we have a bilinear form on \(K_0(A)\) given by \[ <\dim M,\dim N> = \sum_{i\geq 0} (-1)^i\dim_k\mathrm{Ext}^i(M,N). \] The Euler characteristic of \(A\) is the quadratic form \(\chi_A\) associated to \(<-,->\). An element \(x\) of \(K_0(A)\) is called a root of \(\chi_A\) provided \(\chi_A(x)=1\) and, in case \(\chi_A\) is semidefinite, \(x\) is called a radical vector provided \(\chi_A(x)=0\). Finally we say that the category \(A\)-mod of finite dimensional left \(A\)-modules is controlled by \(\chi_A\) provided the following conditions are satisfied: (1) for any indecomposable module \(M\) in \(A\)-mod, \(\dim M\) is either a connected positive root or a connected positive radical vector of \(\chi_A\); (2) for any connected positive root \(x\) of \(\chi_A\) there is precisely one isomorphism class of indecomposable \(A\)-modules \(M\) with \(\dim M=x\); (3) for any connected positive radical vector \(x\) of \(\ch…
Mathematics Subject Classification ID: 16Gxx
Mathematics Subject Classification ID: 16-02
Mathematics Subject Classification ID: 16P10
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 16E20
Mathematics Subject Classification ID: 15A63
Mathematics Subject Classification ID: 11E04
zbMATH DE Number: 3869545
zbMATH Keywords: tilted algebras
zbMATH Keywords: representation theory of finite dimensional algebras
zbMATH Keywords: tame algebras of finite global dimension
zbMATH Keywords: indecomposable modules
zbMATH Keywords: quadratic forms
zbMATH Keywords: basic finite dimensional K-algebra
zbMATH Keywords: simple A-modules
zbMATH Keywords: composition series
zbMATH Keywords: Grothendieck group
zbMATH Keywords: Euler characteristic
zbMATH Keywords: connected positive root
zbMATH Keywords: connected positive radical vector
zbMATH Keywords: tame concealed algebras
zbMATH Keywords: domestic tubular extensions
zbMATH Keywords: tubular algebras
zbMATH Keywords: vector space categories
zbMATH Keywords: integral quadratic forms
zbMATH Keywords: representation-finite algebras
reviewed by: Andrzej Skowroński
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bfb0072870
OpenAlex ID: W1548384067
DOI: 10.1007/BFB0072870
description: scientific article; zbMATH DE number 3869545