On neutral Tannakian subcategories of loop quiver representations
type: Item
instance of: scholarly article
title: On neutral Tannakian subcategories of loop quiver representations
author: Umesh V. Dubey
author: Q6129262
published in: Bulletin des Sciences Mathématiques
publication date: 2024-08-23T00:00:00Z
review text: Tannakian categories appear when studying the cohomology theory of algebraic varieties [\textit{N. Saavedra Rivano}, Bull. Soc. Math. Fr. 100, 417--430 (1972; Zbl 0246.14003)]. Following [\textit{M. V. Nori}, Proc. Indian Acad. Sci., Math. Sci. 91, 73--122 (1982; Zbl 0586.14006)], they can be defined as \(k\)-linear abelian symmetric monoidal categories satisfying eight properties. Additionally, the authors of the reviewed article define semi-Tannakian categories as those that satisfy the first seven conditions but not necessarily the eighth one, which is known as rigidity.\N\NGiven a quiver \(Q\), an ideal \(I\) in the path algebra \(kQ\) and a \(k\)-category \(\mathcal{C}\), one can study the category of representations \(\mathbf{Rep}(Q/I,\mathcal{C})\). In particular for \(Q= L_m\), which is the quiver consisting of a vertex and \(m\) arrows, \(\mathbf{Rep}(L_m /I,\mathcal{C})\) is referred as a category of loop quiver representations. It is well known that the category \(\mathbf{Rep}(L_m /I,\mathcal{C})\) inherits many of the properties of \(\mathcal{C}\). The objective of the authors was to study under what conditions the property of being Tannakian or semi-Tannakian is also inherited. For this purpose, they study two possible tensors in the category \(\mathbf{Rep}(L_m /I,\mathcal{C})\). The first, called the Kronecker tensor, is defined as \((C,\phi_i) \otimes (C', \phi_i '): =(C'\otimes C', \phi_i \otimes \phi_i ')\); and the second, called Simpson tensor, is defined as \((C,\phi_i) \boxtimes (C ', \phi_i '): =(C'\otimes C', \phi_i \otimes 1 + 1 \otimes \phi_i ')\). The main results show that the property of being semi-Tannakian is always inherited, and conditions are given for the condition of being Tannakian to be inherited.\N\NHaving solved the above problem, the authors study and present answers for the analogous problem for the categories of essentially finite loop quiver representations, the category of loop quiver bundles (vector bundles over a field o…
reviewed by: Alejandro Argudín Monroy
Mathematics Subject Classification ID: 18M25
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 14H60
Mathematics Subject Classification ID: 18M05
Mathematics Subject Classification ID: 14F35
zbMATH DE Number: 7900731
zbMATH Keywords: quiver representation
zbMATH Keywords: Tannakian category
zbMATH Keywords: vector bundles
MaRDI profile type: Publication
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DOI: 10.1016/J.BULSCI.2024.103484
description: scientific article; zbMATH DE number 7900731