A Universal Investigation of $n$-representations of $n$-quivers

Document Type : Research Paper


Mathematics Department, College of Computer Sciences and Mathematics, University of Thi-Qar, Iraq



\noindent We have two goals in this paper. First, we investigate and construct cofree coalgebras over $n$-representations of quivers, limits and colimits of $n$-representations of quivers, and limits and colimits of coalgebras in the monoidal categories of $n$-representations of quivers. Second, for any given quivers $\mathit{Q}_1$,$\mathit{Q}_2$,..., $\mathit{Q}_n$, we construct a new quiver $\mathscr{Q}_{\!_{(\mathit{Q}_1, \mathit{Q}_2,..., \mathit{Q}_n)}}$, called an $n$-quiver, and identify each category $Rep_k(\mathit{Q}_j)$ of representations of a quiver $\mathit{Q}_j$ as a full subcategory of the category $Rep_k(\mathscr{Q}_{\!_{(\mathit{Q}_1, \mathit{Q}_2,..., \mathit{Q}_n)}})$ of representations of $\mathscr{Q}_{\!_{(\mathit{Q}_1, \mathit{Q}_2,..., \mathit{Q}_n)}}$ for every $j \in \{1,2,\ldots , n\}$.


[1] Abdulwahid, A.H. and Iovanov, M.C., Generators for comonoids and universal constructions, Arch. Math. 106 (2016), 21-33.
[2] Adamek, J., Herrlich, H., and Strecker, G.E., “Abstract and Concrete Categories: The Joy of Cats”, Dover Publication, 2009.
[3] Assem, I., Skowronski, A., and Simson, D., “Elements of the Representation Theoryof Associative Algebras 1: Techniques of Representation Theory”, London Math. Soc.Student Texts 65, Cambridge University Press, 2006.
[4] Auslander, M., Reiten, I., S.O. Smalø, S.O., “Representation Theory of Artin Algebras”,Cambridge Studies in Advanced Mathematics 36, Cambridge University Press,1995.
[5] Awodey, S., “Category Theory”, Oxford University Press, 2010.
[6] Bakalov, B. and Kirillov, A., Jr., “Lectures on Tensor Categories and Modular Functor”,University Lecture Series 21, American Math. Soc., 2001.
[7] Barot, M., “Introduction to the Representation Theory of Algebras”, Springer, 2015.
[8] Benson, D.J., “Representations and Cohomology I: Basic Representation Theory ofFinite Groups and Associative Algebras”, Cambridge Stud. Adv. Math. 30, CambridgeUniversity Press, 1991.
[9] Borceux, F., “Handbook of Categorical Algebra 1: Basic Category Theory”, CambridgeUniversity Press, 1994.
[10] Borceux, F., “Handbook of Categorical Algebra 2: Categories and Structures”, CambridgeUniversity Press, 1994.
[11] Buan, A.B., Reiten, I., and Solberg, O., “Algebras, Quivers and Representations”,Springer, 2013.
[12] Dvascvalescu, S., Iovanov, M., Nvastvasescu, C., Quiver algebras, path coalgebras andco-reflexivity, Pacific J. Math. 262 (2013), 49-79.
[13] Etingof, P., Gelaki, S., Nikshych, D., and Ostrik, V., “Tensor Categories”. MathematicalSurveys and Monographs 205, American Math. Soc., 2015.
[14] Etingof, P., Golberg, O., Hensel, S., Liu, T., Schwendner, A., Vaintrob, D., Yudovina,E., “Introduction to Representation Theory”, Student Mathematical Library 59,American Math. Soc., 2011.
[15] Freyd, P.J. and Scedrov, A., “Categories, Allegories”, Elsevier Science PublishingCompany, 1990.
[16] Gabriel, P., Unzerlegbare Darstellungen I, Manuscripta Math. 6 (1972), 71-103.
[17] Kilp, M., Knauer, U., Mikhalev, A.V., “Monoids, Acts, and Categories: With Applicationsto Wreath Products and Graphs”, De Gruyter exposition in Math. 29, 2000.
[18] Leinster, T., “Basic Category Theory”, London Math. Soc. Lecture Note Series 298,Cambridge University Press, 2014.
[19] Leinster, T., “Higher Operads, Higher Categories”, Lecture Note Series 298, LondonMath. Soc., Cambridge University Press, 2004.
[20] Mac Lane, S., “Categories for the Working Mathematician”, Graduate Texts in Math.5, Springer-Verlag, 1998.
[21] McLarty, C., “Elementary Categories, Elementary Toposes”, Oxford University Press,2005.
[22] Mitchell, B., “Theory of Categories”, Academic Press, 1965.
[23] Pareigis, B., “Categories and Functors”, Academic Press, 1971.
[24] Rotman, J.J., “An Introduction to Homological Algebra”, Springer, 2009.
[25] Schiffler, R., “Quiver Representations”, CMS Books in Math. Series, Springer InternationalPublishing, 2014.
[26] Schubert, H., “Categories”, Springer-Verlag, 1972.
[27] Sergeichuk, V.V., Linearization method in classification problems of linear algebra,S~ao Paulo J. Math. Sci. 1(2) (2007), 219-240.
[28] Street, R., “Quantum Groups: a Path to Current Algebra”, Lecture Series 19, AustralianMath. Soc., Cambridge University Press, 2007.
[29] Wisbauer, R., “Foundations of Module and Ring Theory: A Handbook for Study andResearch”, Springer-Verlag, 1991.
[30] Zimmermann, A., “Representation Theory: A Homological Algebra Point of View”,Springer-Verlag, 2014.