On neutral Tannakian subcategories of loop quiver representations (Q6592014)

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 on complete connected and geometrically reduced schemes), and the category of twisted loop quiver bundles with relations.