Wild canonical algebras
type: Item
instance of: scholarly article
title: Wild canonical algebras
zbMATH Open document ID: 0882.16011
DOI: 10.1007/PL00004589
author: Helmut Lenzing
author: José A. de la Peña
published in: Mathematische Zeitschrift
publication date: 1997-07-03T00:00:00Z
review text: Canonical algebras had been introduced by \textit{C. M. Ringel} [Tame algebras and integral quadratic forms, Lect. Notes Math. 1099 (1984; Zbl 0546.16013)]. Let \(k\) be an algebraically closed field, \(t\geq 2\) a natural number, \(p=(p_1,\dots,p_t)\) a sequence of natural numbers with \(p_i\geq 2\) and \(\lambda=(\lambda_3,\dots,\lambda_t)\) a sequence of pairwise different nonzero elements in \(k\). Let \({\mathcal Q}\) be the quiver \[ \begin{matrix} & &\overset{\vec x_1}\circ &\overset {x_1}\longrightarrow &\overset {2\vec x_1}\circ &\longrightarrow &\cdots &\longrightarrow &\overset{(p_1-1)\vec x_1}\circ \\ &\overset {x_1}\nearrow &&&&&&&& \overset {x_1}\searrow \\ \overset {0}\circ &\underset {x_2}\longrightarrow &\overset{\vec x_2}\circ & \underset {x_2}\longrightarrow &\overset{2\vec x_2}\circ &\longrightarrow &\cdots &\longrightarrow &\overset{(p_2-1)\vec x_2}\circ &\underset {x_2}\longrightarrow &\overset\vec {c}\circ \\ &\underset {x_t}\searrow &&\vdots &&\vdots &&\vdots &&\underset {x_t}\nearrow \\ & &\underset {\vec x_t}\circ &\underset {x_t}\longrightarrow &\underset{2\vec x_t}\circ &\longrightarrow &\cdots &\longrightarrow &\underset {(p_t-1)\vec x_t}\circ\end{matrix} \] The canonical algebra \(\Lambda=\Lambda(p,\lambda)\) is the path-algebra of the quiver \({\mathcal Q}\), bound by the relations \(x^{p_i}_i-x_2^{p_2}+\lambda_i x_1^{p_1}\). If the tree \({\mathcal T}\), obtained from \({\mathcal Q}\) by deleting the vertex 0 is a Dynkin diagram, then \(\Lambda\) is a well known tame concealed algebra. If \({\mathcal T}\) is an extended Dynkin diagram, then \(\Lambda\) is a tubular algebra. Ringel [loc. cit.] gave a description of the category \(\text{mod }\Lambda\) of finite dimensional \(\Lambda\)-modules in this case. The paper under review studies the case, when \({\mathcal T}\) is a wild quiver and \(\Lambda\) then is called wild canonical. In this case, the module category \(\text{mod }\Lambda\) decomposes into three parts, \(\text{mod }\Lambda=…
Mathematics Subject Classification ID: 16G70
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 16G60
Mathematics Subject Classification ID: 16D90
zbMATH DE Number: 1028150
zbMATH Keywords: categories of finite dimensional modules
zbMATH Keywords: hereditary categories of coherent sheaves
zbMATH Keywords: canonical algebras
zbMATH Keywords: path-algebras
zbMATH Keywords: quivers
zbMATH Keywords: tame concealed algebras
zbMATH Keywords: extended Dynkin diagrams
zbMATH Keywords: tubular algebras
zbMATH Keywords: wild quivers
zbMATH Keywords: Auslander-Reiten quivers
zbMATH Keywords: separating families
zbMATH Keywords: orthogonal standard tubes
zbMATH Keywords: preprojective components
zbMATH Keywords: indecomposable projectives
zbMATH Keywords: Auslander-Reiten components
zbMATH Keywords: quasitilted algebras
zbMATH Keywords: weighted projective lines
zbMATH Keywords: tilting vector bundles
zbMATH Keywords: minimal projective generators
zbMATH Keywords: right perpendicular categories
zbMATH Keywords: endomorphism rings
zbMATH Keywords: categories of finite length sheaves
zbMATH Keywords: relative Auslander-Reiten translations
zbMATH Keywords: wild tilted algebras
zbMATH Keywords: dimension vectors
reviewed by: Otto Kerner
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1028150