Constructions of n-cluster tilting subcategories using representation-directed algebras
Abstract: One of the most useful tools in representation theory of algebras is Auslander–Reiten theory. A higher dimensional analogue has recently appeared, based on the notion of n-cluster tilting subcategories. It turns out that the existence of such subcategories in the module category of an algebra gives important information about the whole module category. However it is unclear in general which algebras have module categories that admit an n-cluster tilting subcategory. This thesis provides many different ways to construct n-cluster tilting subcategories. The main tools are representation-directed algebras, methods from classical Auslander–Reiten theory and homological algebra.In Paper I we give a characterization of n-cluster tilting subcategories for representation-directed algebras. Using this characterization, we classify acyclic Nakayama algebras with homogeneous relations whose module categories admit an n-cluster tilting subcategory. We provide a formula for the global dimension of such algebras and then classify all Nakayama algebras with homogeneous relations whose module categories admit a d-cluster tilting subcategory, where d is the global dimension of the algebra.In Paper II we construct a generalization of n-cluster tilting subcategories for representation-directed algebras called n-fractured subcategories. By defining a gluing procedure we are able to construct n-cluster tilting subcategories for representation-directed algebras by gluing compatible n-fractured subcategories. As an application of this method, for any positive integer n and any positive integer d ≥ 2n we explicitly construct an algebra with global dimension d whose module category admits an n-cluster tilting subcategory. If n is odd, then this result can be improved to any d ≥ n.In Paper III we further generalize the gluing procedure defined in Paper II. First we give sense to a notion of infinite gluings which gives an n-cluster tilting subcategory for an abelian category which is not the module category of an algebra. Next, we apply an orbit construction to obtain an algebra which is not necessarily representation-directed but its module category admits an n-cluster tilting subcategory. Finally we classify starlike algebras with radical square zero relations whose module categories admit n-cluster tilting subcategories. Using the theory developed in this paper, we present many families of algebras whose module categories admit n-cluster tilting subcategories.
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