Semi-invariants of canonical algebras
type: Item
instance of: scholarly article
title: Semi-invariants of canonical algebras
zbMATH Open document ID: 0947.16004
author: Jerzy Weyman
author: Andrzej Skowroński
published in: Manuscripta Mathematica
publication date: 2000-02-08T00:00:00Z
review text: Let \(K\) be an algebraically closed field, and let \(A\) be a finite dimensional associative \(K\)-algebra with an identity which is assumed to be basic and connected. There is an isomorphism \(A\cong KQ/I\), where \(KQ\) is the path algebra of the Gabriel quiver \(Q=Q_A\) of \(A\) and \(I\) is an admissible ideal of \(KQ\) generated by a finite system of forms (so called \(K\)-linear relations). Let \(n\geq 3\), \({\mathbf m}=(m_1,\ldots,m_n)\in\mathbb{Z}^n\), \(m_i\geq 2\), \({\mathbf x}=(x_3,\ldots,x_n)\in(K\setminus 0)^{n-2}\), \(x_3=1\), \(x_i\neq x_j\) for all \(i\neq j\). Denote by \(\Lambda({\mathbf m},{\mathbf x})=K\Delta(m_1,\ldots,m_n)/I(x_3,\ldots,x_n)\) the canonical algebra associated with the pair \(({\mathbf m},{\mathbf x})\) [\textit{C. M. Ringel}, Tame algebras and integral quadratic forms. Lect. Notes Math. 1099 (1984; Zbl 0546.16013)]. The paper is devoted to the study of the algebras of semi-invariants on the varieties of regular representations of canonical algebras \(\Lambda({\mathbf m},{\mathbf x})\). In particular, when \(\text{char }K=0\) the authors obtain an explicit description of such algebras and prove that these algebras are complete intersections or polynomial algebras. The proof is based on the techniques developed in their forthcoming paper [\textit{A. Skowroński, J. Weyman}, The algebras of semi-invariants of quivers. Transform. Groups (to appear)].
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 14M10
Mathematics Subject Classification ID: 16G60
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 16G70
zbMATH DE Number: 1385141
zbMATH Keywords: finite dimensional algebras
zbMATH Keywords: Gabriel quivers
zbMATH Keywords: Kronecker quivers
zbMATH Keywords: path algebras
zbMATH Keywords: canonical algebras
zbMATH Keywords: regular representations
zbMATH Keywords: semi-invariants
zbMATH Keywords: complete intersections
zbMATH Keywords: regular modules
reviewed by: Aleksandr G. Aleksandrov
MaRDI profile type: Publication
full work available at URL: s002290050208
OpenAlex ID: W1986120813
DOI: 10.1007/S002290050208
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description: scientific article; zbMATH DE number 1385141