Posets of finite prinjective type and a class of orders
type: Item
instance of: scholarly article
title: Posets of finite prinjective type and a class of orders
zbMATH Open document ID: 0815.16006
DOI: 10.1016/0022-4049(93)90138-J
author: Daniel Simson
published in: Journal of Pure and Applied Algebra
publication date: 1995-06-30T00:00:00Z
review text: Let \(K\) be a field and \(J\) a finite partially ordered set with \(n\) maximal elements, called peaks. The author defines the category of socle projective representations of \(J\) over \(K\) (or peak \(J\)-spaces) generalizing the notion of \(I\)-spaces in the sense of \textit{P. Gabriel} [Manuscr. Math. 6, 71-103 (1972; Zbl 0232.08001)] and the definition of the author [in Proc. Tsukuba Int. Conf. Representations of finite dimensional algebras, CMS Conf. Proc. 11, 287-298 (1991; Zbl 0797.16015)], where the case \(n = 2\) is considered. The main result is a characterization of those posets \(J\) having only finitely many isomorphism classes of indecomposable socle projective representations. The characterization is given in terms of the Tits quadratic form associated to \(J\), in terms of algebraic varieties with an algebraic group action and by presenting a critical list of 110 minimal posets having infinitely many isomorphism classes of indecomposable socle projective representations. The result generalizes the characterization of representation-finite one peak posets of \textit{M. M. Kleiner} [Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 28, 32-41 (1972; Zbl 0345.06001)] and the author's characterization dealing with the case of two peaks given in the paper mentioned above. One of the methods used in the proof is the study of prinjective \(KJ\)-modules investigated by \textit{J. A. de la Peña} and the author [in Trans. Am. Math. Soc. 329, 733-753 (1992; Zbl 0789.16010)]. An application to the study of lattices over orders is given.
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 16G30
Mathematics Subject Classification ID: 06A06
Mathematics Subject Classification ID: 16G60
zbMATH DE Number: 500601
zbMATH Keywords: finite partially ordered sets
zbMATH Keywords: category of socle projective representations
zbMATH Keywords: indecomposable socle projective representations
zbMATH Keywords: Tits quadratic form
zbMATH Keywords: minimal posets
zbMATH Keywords: representation-finite posets
zbMATH Keywords: lattices over orders
reviewed by: Hagen Meltzer
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 500601