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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.
nrepresentation infinite algebras
"... From the viewpoint of higher dimensional AuslanderReiten theory, we introduce a new class of finite dimensional algebras of global dimension n, which we call nrepresentation infinite. They are a certain analog of representation infinite hereditary algebras, and we study three important classes o ..."
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From the viewpoint of higher dimensional AuslanderReiten theory, we introduce a new class of finite dimensional algebras of global dimension n, which we call nrepresentation infinite. They are a certain analog of representation infinite hereditary algebras, and we study three important classes of modules: npreprojective, npreinjective and nregular modules. We observe that their homological behaviour is quite interesting. For instance they provide first examples of algebras having infinite Ext 1orthogonal families of modules. Moreover we give general constructions of nrepresentation infinite algebras. Applying Minamoto’s theory on Fano algebras in noncommutative algebraic geometry, we describe the category of nregular modules in terms of the corresponding preprojective algebra. Then we introduce nrepresentation tame algebras, and show that the category of nregular modules decomposes into the categories of finite dimensional modules over localizations of the preprojective algebra. This generalizes the classical description of regular modules over tame hereditary algebras. As an application, we show that the representation dimension of an nrepresentation tame algebra is at least n+2.
HIGHER PREPROJECTIVE ALGEBRAS AND STABLY CALABIYAU PROPERTIES
, 2013
"... Abstract. In this paper, we give sufficient properties for a finite dimensional graded algebra to be a higher preprojective algebra. These properties are of homological nature, they use Gorensteiness and bimodule isomorphisms in the stable category of CohenMacaulay modules. We prove that these prop ..."
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Abstract. In this paper, we give sufficient properties for a finite dimensional graded algebra to be a higher preprojective algebra. These properties are of homological nature, they use Gorensteiness and bimodule isomorphisms in the stable category of CohenMacaulay modules. We prove that these properties are also necessary for 3preprojective algebras using [Kel11] and for preprojective algebras of higher representation finite algebras using [Dug12].
From Auslander algebras to tilted algebras
"... For an (n − 1)Auslander algebra Λ with global dimension n ≥ 2, we show that if Λ admits a trivial maximal (n − 1)orthogonal subcategory of mod Λ, then Λ is of finite representation type and the projective dimension or injective dimension of any indecomposable module in mod Λ is at most n − 1. As a ..."
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For an (n − 1)Auslander algebra Λ with global dimension n ≥ 2, we show that if Λ admits a trivial maximal (n − 1)orthogonal subcategory of mod Λ, then Λ is of finite representation type and the projective dimension or injective dimension of any indecomposable module in mod Λ is at most n − 1. As a result, we have that for an Auslander algebra Λ with global dimension 2, if Λ admits a trivial maximal 1orthogonal subcategory of mod Λ, then Λ is a tilted algebra of finite representation type; furthermore, in case there exists a unique simple module in mod Λ with projective dimension 2, then the converse also holds true. 1.
Trivial Maximal 1Orthogonal Subcategories For Auslander’s 1Gorenstein Algebras
, 2009
"... Let Λ be an Auslander’s 1Gorenstein Artinian algebra with global dimension 2. If Λ admits a trivial maximal 1orthogonal subcategory of mod Λ, then for any indecomposable module M ∈ mod Λ, we have that the projective dimension of M is equal to 1 if and only if so is its injective dimension and th ..."
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Let Λ be an Auslander’s 1Gorenstein Artinian algebra with global dimension 2. If Λ admits a trivial maximal 1orthogonal subcategory of mod Λ, then for any indecomposable module M ∈ mod Λ, we have that the projective dimension of M is equal to 1 if and only if so is its injective dimension and that M is injective if the projective dimension of M is equal to 2, which implies that Λ is almost hereditary.
nREPRESENTATIONFINITE ALGEBRAS AND FRACTIONALLY CALABIYAU ALGEBRAS
, 2009
"... In this short paper, we study nrepresentationfinite algebras from the viewpoint of fractionally CalabiYau algebras. We shall show that all nrepresentationfinite algebras are twisted fractionally CalabiYau. We also show that twisted n(ℓ−1)CalabiYau algebras of global dimension n are ℓ nrepr ..."
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In this short paper, we study nrepresentationfinite algebras from the viewpoint of fractionally CalabiYau algebras. We shall show that all nrepresentationfinite algebras are twisted fractionally CalabiYau. We also show that twisted n(ℓ−1)CalabiYau algebras of global dimension n are ℓ nrepresentationfinite for any ℓ> 0. As an application, we give a construction of nrepresentationfinite algebras using the tensor product.
Trivial extensions, iterated tilted algebras and clustertilted algebras
 S~AO PAULO JOURNAL OF MATHEMATICAL SCIENCES 4, 3 (2010), 499{527
, 2010
"... Classical results in representation theory establish very interesting connections between iterated tilted algebras and trivial extensions of finite dimensional algebras, particularly deep and useful in the Dynkin case. Recent results show that there are also interesting connections ..."
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Classical results in representation theory establish very interesting connections between iterated tilted algebras and trivial extensions of finite dimensional algebras, particularly deep and useful in the Dynkin case. Recent results show that there are also interesting connections
Derived equivalences in nangulated categories
 Algebr. Represent. Theory
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