Results 1 
9 of
9
nrepresentationfinite algebras and nAPR tilting
 6575–6614 (2011) Zbl pre05987996 MR 2833569
"... Abstract. We introduce the notion of nrepresentationfiniteness, generalizing representationfinite hereditary algebras. We establish the procedure of nAPR tilting and show that it preserves nrepresentationfiniteness. We give some combinatorial description of this procedure and use this to compl ..."
Abstract

Cited by 21 (11 self)
 Add to MetaCart
Abstract. We introduce the notion of nrepresentationfiniteness, generalizing representationfinite hereditary algebras. We establish the procedure of nAPR tilting and show that it preserves nrepresentationfiniteness. We give some combinatorial description of this procedure and use this to completely describe a class of nrepresentationfinite algebras called “type A”. Contents
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 ..."
Abstract

Cited by 6 (2 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
(Show Context)
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.
τ2stable tilting complexes over weighted projective lines. arXiv:1402.6036
, 2014
"... Abstract. Let X be a weighted projective line and cohX the associated categoy of coherent sheaves. We classify the tilting complexes T in Db(cohX) such that τ2T ∼ = T, where τ is the AuslanderReiten translation in Db(cohX). As an application of this result, we classify the 2representationfinite ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
Abstract. Let X be a weighted projective line and cohX the associated categoy of coherent sheaves. We classify the tilting complexes T in Db(cohX) such that τ2T ∼ = T, where τ is the AuslanderReiten translation in Db(cohX). As an application of this result, we classify the 2representationfinite algebras which are derivedequivalent to a canonical algebra. This complements IyamaOppermann’s classification of the iterated tilted 2representationfinite algebras. By passing to 3preprojective algebras, we obtain a classification of the selfinjective clustertilted algebras of canonicaltype. This complements Ringel’s classification of the selfinjective clustertilted algebras. 1.
doi:10.1112/jlms/jds034
"... On derived equivalences of lines, rectangles and triangles ..."
(Show Context)
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 ..."
Abstract
 Add to MetaCart
(Show Context)
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