One-parameter equipped posets and their representations
type: Item
instance of: scholarly article
title: One-parameter equipped posets and their representations
zbMATH Open document ID: 0970.16010
DOI: 10.1007/BF02482429
author: A. V. Zabarilo
author: Alexander Zavadskij
publication date: 2001-03-27T00:00:00Z
review text: Representations of equipped posets can be interpreted in terms of matrix problems over the field \(\mathbb{R}\) of real numbers corresponding to a special type of vector space categories over \(\mathbb{R}\). In particular the considered vector space categories have indecomposable objects with endomorphism ring \(\mathbb{R}\) or \(\mathbb{C}\) (the field of complex numbers) only. The possible bimodules of morphisms between indecomposable objects are \(\mathbb{R}\), \(\mathbb{C}\) and \(\left(\begin{smallmatrix}\mathbb{R} &\mathbb{R}\\ \mathbb{R} &\mathbb{R}\end{smallmatrix}\right)\). An equipped poset is a partially ordered set with an additional structure encoding the corresponding matrix problem. The paper announces the description of equipped posets which are of infinite representation type and (domestic) one-parameter in the sense analogous to the usual one [see Section 15.10 in \textit{D. Simson}, ``Linear representations of partially ordered sets and vector space categories'', Algebra, Logic and Applications, Vol. 4, Gordon \& Breach Science Publishers, New York (1992; Zbl 0818.16009)]. The combinatorial criterion for an equipped poset to be one-parameter generalizes the result of \textit{V. V. Otrashevskaya} [in: Matrix problems, Kiev 1977, 144-149 (1977; Zbl 0461.16025)]. The possible coordinate vectors of indecomposable representations of one-parameter equipped posets are described in terms of roots and imaginary roots of the associated Tits quadratic form and so-called special vectors. Also a classification of sincere nontrivially equipped posets is given, cf. \textit{A. G. Zavadskij} [Inst. Mat. Akad. Nauk Ukr. SSR, Kiev 1981, Preprint 81.27, 30-42 (1981)].
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 16G60
zbMATH DE Number: 1505731
zbMATH Keywords: equipped posets
zbMATH Keywords: matrix problems
zbMATH Keywords: one-parameter families of representations
zbMATH Keywords: domestic sincere posets
zbMATH Keywords: vector space categories
zbMATH Keywords: infinite representation type
zbMATH Keywords: Tits quadratic forms
reviewed by: Stanisław Kasjan
MaRDI profile type: Publication
cites work: Algebra VIII: Representations of finite-dimensional algebras. Transl. from the Russian
cites work: Schurian sp-representation-finite right peak PI-rings and their indecomposable socle projective modules
cites work: Normal forms of real matrices with respect to complex similarity
Recommended article: Representations of one-parameter equipped posets. II
Recommended article: Representations of one-parameter equipped posets
Recommended article: On corepresentations of one parametric equipped posets.
Recommended article: ON THE GABRIEL’S QUIVER OF SOME EQUIPPED POSETS
Recommended article: Q5701961
Recommended article: Algebraically equipped posets
Recommended article: Tame equipped posets.
Recommended article: Morphisms in categories of representations of equipped posets.
Recommended article: Q3145827
Recommended article: Q4011899
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 1505731