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In the computation of some representation-theoretic numerical invariants of domestic string algebras, a finite combinatorial gadget introduced by Schr{o}er--the emph{bridge quiver} whose vertices are (representatives of cyclic permutations of) bands and whose arrows are certain band-free strings--has been used extensively. There is a natural but ill-behaved partial binary operation, $circ$, on the larger set of emph{weak bridges} such that bridges are precisely the $circ$-irreducibles. With the goal of computing hammocks up to isomorphism in a later work we equip an even larger set of emph{weak arch bridges} with another partial binary operation, $circ_H$, to obtain a finite category. Each weak arch bridge admits a unique $circ_H$-factorization into emph{arch bridges}, i.e., the $circ_H$-irreducibles.
We provide the localization procedure for monoidal categories by a real commuting family of braiders. For an element $w$ of the Weyl group, $mathscr{C}_w$ is a subcategory of modules over quiver Hecke algebra which categorifies the quantum unipotent
Following [20], a desingularization of arbitrary quiver Grassmannians for finite dimensional Gorenstein projective modules of 1-Gorenstein gentle algebras is constructed in terms of quiver Grassmannians for their Cohen-Macaulay Auslander algebras.
This is an introduction to some aspects of Fomin-Zelevinskys cluster algebras and their links with the representation theory of quivers and with Calabi-Yau triangulated categories. It is based on lectures given by the author at summer schools held in
In this paper we develop combinatorial techniques for the case of string algebras with the aim to give a characterization of string complexes with infinite minimal projective resolution. These complexes will be called textit{periodic string complexes
Let $U_q(mathfrak{g})$ be a twisted affine quantum group of type $A_{N}^{(2)}$ or $D_{N}^{(2)}$ and let $mathfrak{g}_{0}$ be the finite-dimensional simple Lie algebra of type $A_{N}$ or $D_{N}$. For a Dynkin quiver of type $mathfrak{g}_{0}$, we defin