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Higher dimensional cluster combinatorics and representation theory
"... Higher Auslander algebras were introduced by Iyama generalizing classical concepts from representation theory of finite dimensional algebras. Recently these higher analogues of classical representation theory have been increasingly studied. Cyclic polytopes are classical objects of study in convex ..."
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Higher Auslander algebras were introduced by Iyama generalizing classical concepts from representation theory of finite dimensional algebras. Recently these higher analogues of classical representation theory have been increasingly studied. Cyclic polytopes are classical objects of study in convex geometry. In particular, their triangulations have been studied with a view towards generalizing the rich combinatorial structure of triangulations of polygons. In this paper, we demonstrate a connection between these two seemingly unrelated subjects. We study triangulations of evendimensional cyclic polytopes and tilting modules for higher Auslander algebras of linearly oriented type A which are summands of the cluster tilting module. We show that such tilting modules correspond bijectively to triangulations. Moreover mutations of tilting modules correspond to bistellar flips of triangulations. For any drepresentation finite algebra we introduce a certain ddimensional cluster category and study its cluster tilting objects. For higher Auslander algebras of linearly oriented type A we obtain a similar correspondence between cluster tilting objects and triangulations of a certain cyclic polytope. Finally we study certain functions on generalized laminations in cyclic polytopes, and show that they satisfy analogues of tropical cluster exchange relations. Moreover we observe that the terms of these exchange relations are closely related to the terms occuring in the mutation of cluster tilting objects.
Derived equivalences in nangulated categories
 Algebr. Represent. Theory
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"... Abstract. Over any field of positive characteristic we construct 2CYtilted algebras that are not Jacobian algebras of quivers with potentials. As a remedy, we propose an extension of the notion of a potential, called hyperpotential, that allows to prove that certain algebras defined over fields o ..."
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Abstract. Over any field of positive characteristic we construct 2CYtilted algebras that are not Jacobian algebras of quivers with potentials. As a remedy, we propose an extension of the notion of a potential, called hyperpotential, that allows to prove that certain algebras defined over fields of positive characteristic are 2CYtilted even if they do not arise from potentials. In another direction, we compute the fractionally CalabiYau dimensions of certain orbit categories of fractionally CY triangulated categories. As an application, we construct a cluster category of type G2. Introduction A 2CYtilted algebra is an endomorphism algebra of a clustertilting object in a 2CalabiYau triangulated category. There are close connections between 2CYtilted algebras and Jacobian algebras of quivers with potentials as introduced by Derksen, Weyman and Zelevinsky On the other hand, by the work of Amiot [2], any finitedimensional Jacobian algebra is 2CYtilted. It is therefore natural to ask whether any 2CYtilted algebra is a Jacobian algebra of a quiver with potential [3, Question 2.20]. The purpose of this note is twofold. First, we provide a negative answer to this question over any field of positive characteristic. Our examples are given by certain selfinjective Nakayama algebras which are also known as truncated cycle algebras. Second, we show that it is actually possible to slightly extend the notion of a potential in order to exclude this kind of examples. Let us explain the motivation behind such extension. Since 2CYtilted algebras have some remarkable homological and structural properties Consider for example the algebra Λ K = K[x]/(x n−1 ) over a field K for some n > 2, which could be described as a quiver with one vertex, one loop x and a relation x n−1 . Date: March 26, 2014.
TORSION CLASSES AND tSTRUCTURES IN HIGHER HOMOLOGICAL ALGEBRA
"... Abstract. Higher homological algebra was introduced by Iyama. It is also known as nhomological algebra where n> 2 is a fixed integer, and it deals with ncluster tilting subcategories of abelian categories. All short exact sequences in such a subcategory are split, but it has nice exact sequence ..."
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Abstract. Higher homological algebra was introduced by Iyama. It is also known as nhomological algebra where n> 2 is a fixed integer, and it deals with ncluster tilting subcategories of abelian categories. All short exact sequences in such a subcategory are split, but it has nice exact sequences with n + 2 objects. This was recently formalised by Jasso in the theory of nabelian categories. There is also a derived version of nhomological algebra, formalised by Geiss, Keller, and Oppermann in the theory of (n+ 2)angulated categories (the reason for the shift from n to n + 2 is that angulated categories have triangulated categories as the “base case”). We introduce torsion classes and tstructures into the theory of nabelian and (n + 2)angulated categories, and prove several results to motivate the definitions. Most of the results concern the nabelian and (n+2)angulated categoriesM (Λ) and C (Λ) associated to an nrepresentation finite algebra Λ, as defined by Iyama and Oppermann. We characterise torsion classes in these categories in terms of closure under higher extensions, and give a bijection between torsion classes in M (Λ) and intermediate tstructures in C (Λ) which is a category one can reasonably view as the nderived category of M (Λ). We hint at the link