Supercanonical algebras.
type: Item
instance of: scholarly article
title: Supercanonical algebras.
zbMATH Open document ID: 1092.16007
author: Helmut Lenzing
author: José A. de la Peña
published in: Journal of Algebra
publication date: 2004-11-29T00:00:00Z
review text: The authors introduce a new family of algebras known as supercanonical algebras. Up to derived equivalence, these algebras include the tame hereditary algebras, the tubular algebras, the canonical algebras, the concealed-canonical algebras and the quasitilted algebras of canonical type. Supercanonical algebras, while not necessarily possessing families of stable tubes which are separating or without external short cycles (i.e. certain nice properties), still have families of stable tubes with good properties. For more on canonical algebras, see \textit{W. Geigle} and \textit{H. Lenzing} [Lect. Notes Math. 1273, 265-297 (1987; Zbl 0651.14006)], \textit{C. M. Ringel} [Tame algebras and integral quadratic forms, (Lect. Notes Math. 1099) (1984; Zbl 0546.16013)]. The data required for the definition is an algebraically closed field \(k\), finite posets \(S_1,S_2,\dots,S_t\) and distinct non-zero elements \(\lambda_3,\lambda_4,\dots,\lambda_t\in k\) (\(t\geq 2\)). Let \(\widetilde S\) denote the double cone of any finite poset \(S\) (obtained by adding maximum and minimum elements to \(S\)). Form a new poset by identifying the maximum and minimum elements of all the double cones \(\widetilde{S_i}\). The supercanonical algebra is the corresponding quiver algebra subject to certain relations. The paper forms an extensive study of supercanonical algebras, including their relationship with known families of algebras, their Hochschild cohomology, the Grothendieck group, Euler form, Coxeter matrix, families of tubes, existence of preprojective components, and some of their derived properties.
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 16G70
Mathematics Subject Classification ID: 19A49
Mathematics Subject Classification ID: 18F30
Mathematics Subject Classification ID: 16G10
Mathematics Subject Classification ID: 16G60
zbMATH DE Number: 2119566
zbMATH Keywords: representations of algebras
zbMATH Keywords: canonical algebras
zbMATH Keywords: tame hereditary algebras
zbMATH Keywords: separating families
zbMATH Keywords: posets
zbMATH Keywords: stable tubes
zbMATH Keywords: tubular algebras
zbMATH Keywords: concealed-canonical algebras
zbMATH Keywords: quasitilted algebras of canonical type
zbMATH Keywords: Auslander-Reiten translations
zbMATH Keywords: derived categories
zbMATH Keywords: Euler forms
zbMATH Keywords: Hochschild cohomology
zbMATH Keywords: Coxeter matrices
zbMATH Keywords: one-parameter families
zbMATH Keywords: supercanonical algebras
reviewed by: Bethany R. Marsh
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2004.06.021
OpenAlex ID: W4205934134
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DOI: 10.1016/J.JALGEBRA.2004.06.021
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description: scientific article; zbMATH DE number 2119566