Three Representation Types for Systems of Forms and Linear Maps

Alazemi, Abdullah; Anđelić, Milica; da Fonseca, Carlos M.; Futorny, Vyacheslav; Sergeichuk, Vladimir V.

doi:10.3390/math9050455

Open AccessArticle

Three Representation Types for Systems of Forms and Linear Maps

by

Abdullah Alazemi

¹

,

Milica Anđelić

^1,*

,

Carlos M. da Fonseca

^2,3

,

Vyacheslav Futorny

⁴

and

Vladimir V. Sergeichuk

⁵

¹

Department of Mathematics, Kuwait University, Safat 13060, Kuwait

²

Department of Mathematics, Kuwait College of Science and Technology, Safat 13133, Kuwait

³

Chair of Computational Mathematics, University of Deusto, 48007 Bilbao, Spain

⁴

Department of Mathematics, University of São Paulo, São Paulo 05508, Brazil

⁵

Institute of Mathematics, Tereshchenkivska 3, 01024 Kiev, Ukraine

^*

Author to whom correspondence should be addressed.

Mathematics 2021, 9(5), 455; https://doi.org/10.3390/math9050455

Submission received: 6 January 2021 / Revised: 4 February 2021 / Accepted: 19 February 2021 / Published: 24 February 2021

(This article belongs to the Section A: Algebra and Logic)

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Abstract

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.

Keywords: representations of quivers; representations of mixed graphs; wild problems

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MDPI and ACS Style

Alazemi, A.; Anđelić, M.; da Fonseca, C.M.; Futorny, V.; Sergeichuk, V.V. Three Representation Types for Systems of Forms and Linear Maps. Mathematics 2021, 9, 455. https://doi.org/10.3390/math9050455

AMA Style

Alazemi A, Anđelić M, da Fonseca CM, Futorny V, Sergeichuk VV. Three Representation Types for Systems of Forms and Linear Maps. Mathematics. 2021; 9(5):455. https://doi.org/10.3390/math9050455

Chicago/Turabian Style

Alazemi, Abdullah, Milica Anđelić, Carlos M. da Fonseca, Vyacheslav Futorny, and Vladimir V. Sergeichuk. 2021. "Three Representation Types for Systems of Forms and Linear Maps" Mathematics 9, no. 5: 455. https://doi.org/10.3390/math9050455

APA Style

Alazemi, A., Anđelić, M., da Fonseca, C. M., Futorny, V., & Sergeichuk, V. V. (2021). Three Representation Types for Systems of Forms and Linear Maps. Mathematics, 9(5), 455. https://doi.org/10.3390/math9050455

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

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Three Representation Types for Systems of Forms and Linear Maps

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