Three Representation Types for Systems of Forms and Linear Maps

We consider systems of bilinear forms and linear maps as representations of a graph with undirected and directed edges. Its vertices represent vector spaces; its undirected and directed edges represent bilinear forms and linear maps, respectively. We prove that if the problem of classifying represen...

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Main Authors:	Abdullah Alazemi, Milica Anđelić, Carlos M. da Fonseca, Vyacheslav Futorny, Vladimir V. Sergeichuk
Format:	Article
Language:	English
Published:	MDPI AG 2021-02-01
Series:	Mathematics
Subjects:	representations of quivers representations of mixed graphs wild problems
Online Access:	https://www.mdpi.com/2227-7390/9/5/455

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spelling	doaj-9b055c490d1046e89012973d1066f1192021-02-25T00:00:48ZengMDPI AGMathematics2227-73902021-02-01945545510.3390/math9050455Three Representation Types for Systems of Forms and Linear MapsAbdullah Alazemi0Milica Anđelić1Carlos M. da Fonseca2Vyacheslav Futorny3Vladimir V. Sergeichuk4Department of Mathematics, Kuwait University, Safat 13060, KuwaitDepartment of Mathematics, Kuwait University, Safat 13060, KuwaitDepartment of Mathematics, Kuwait College of Science and Technology, Safat 13133, KuwaitDepartment of Mathematics, University of São Paulo, São Paulo 05508, BrazilInstitute of Mathematics, Tereshchenkivska 3, 01024 Kiev, UkraineWe consider systems of bilinear forms and linear maps as representations of a graph with undirected and directed edges. Its vertices represent vector spaces; its undirected and directed edges represent bilinear forms and linear maps, respectively. We prove that if the problem of classifying representations of a graph has not been solved, then it is equivalent to the problem of classifying representations of pairs of linear maps or pairs consisting of a bilinear form and a linear map. Thus, there are only two essentially different unsolved classification problems for systems of forms and linear maps.https://www.mdpi.com/2227-7390/9/5/455representations of quiversrepresentations of mixed graphswild problems
collection	DOAJ
language	English
format	Article
sources	DOAJ
author	Abdullah Alazemi Milica Anđelić Carlos M. da Fonseca Vyacheslav Futorny Vladimir V. Sergeichuk
spellingShingle	Abdullah Alazemi Milica Anđelić Carlos M. da Fonseca Vyacheslav Futorny Vladimir V. Sergeichuk Three Representation Types for Systems of Forms and Linear Maps Mathematics representations of quivers representations of mixed graphs wild problems
author_facet	Abdullah Alazemi Milica Anđelić Carlos M. da Fonseca Vyacheslav Futorny Vladimir V. Sergeichuk
author_sort	Abdullah Alazemi
title	Three Representation Types for Systems of Forms and Linear Maps
title_short	Three Representation Types for Systems of Forms and Linear Maps
title_full	Three Representation Types for Systems of Forms and Linear Maps
title_fullStr	Three Representation Types for Systems of Forms and Linear Maps
title_full_unstemmed	Three Representation Types for Systems of Forms and Linear Maps
title_sort	three representation types for systems of forms and linear maps
publisher	MDPI AG
series	Mathematics
issn	2227-7390
publishDate	2021-02-01
description	We consider systems of bilinear forms and linear maps as representations of a graph with undirected and directed edges. Its vertices represent vector spaces; its undirected and directed edges represent bilinear forms and linear maps, respectively. We prove that if the problem of classifying representations of a graph has not been solved, then it is equivalent to the problem of classifying representations of pairs of linear maps or pairs consisting of a bilinear form and a linear map. Thus, there are only two essentially different unsolved classification problems for systems of forms and linear maps.
topic	representations of quivers representations of mixed graphs wild problems
url	https://www.mdpi.com/2227-7390/9/5/455
work_keys_str_mv	AT abdullahalazemi threerepresentationtypesforsystemsofformsandlinearmaps AT milicaanđelic threerepresentationtypesforsystemsofformsandlinearmaps AT carlosmdafonseca threerepresentationtypesforsystemsofformsandlinearmaps AT vyacheslavfutorny threerepresentationtypesforsystemsofformsandlinearmaps AT vladimirvsergeichuk threerepresentationtypesforsystemsofformsandlinearmaps
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Three Representation Types for Systems of Forms and Linear Maps

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