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Cluster tilting for higher Auslander algebras
, 2008
"... The concept of cluster tilting gives a higher analogue of classical Auslander correspondence between representationfinite algebras and Auslander algebras. The nAuslanderReiten translation functor τn plays an important role in the study of ncluster tilting subcategories. We study the category M ..."
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Cited by 30 (9 self)
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The concept of cluster tilting gives a higher analogue of classical Auslander correspondence between representationfinite algebras and Auslander algebras. The nAuslanderReiten translation functor τn plays an important role in the study of ncluster tilting subcategories. We study the category Mn of preinjectivelike modules obtained by applying τn to injective modules repeatedly. We call a finite dimensional algebra Λ ncomplete if Mn = add M for an ncluster tilting object M. Our main result asserts that the endomorphism algebra EndΛ(M) is (n + 1)complete. This gives an inductive construction of ncomplete algebras. For example, any representationfinite hereditary algebra Λ (1) is 1complete. Hence the Auslander algebra Λ (2) of Λ (1) is 2complete. Moreover, for any n ≥ 1, we have an ncomplete algebra Λ (n) which has an ncluster tilting object M (n) such that Λ (n+1) = End Λ (n)(M (n)). We give the presentation of Λ (n) by a quiver with relations. We apply our results to construct ncluster tilting subcategories of derived categories of ncomplete algebras.
nANGULATED CATEGORIES
"... Abstract. We define nangulated categories by modifying the axioms of triangulated categories in a natural way. We show that Heller’s parametrization of pretriangulations extends to prenangulations. We obtain a large class of examples of nangulated categories by considering (n − 2)cluster tilti ..."
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Cited by 16 (1 self)
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Abstract. We define nangulated categories by modifying the axioms of triangulated categories in a natural way. We show that Heller’s parametrization of pretriangulations extends to prenangulations. We obtain a large class of examples of nangulated categories by considering (n − 2)cluster tilting subcategories of triangulated categories which are stable under the (n−2)nd power of the suspension functor. Finally, as an application, we show how nangulated CalabiYau categories yield triangulated CalabiYau categories of higher CalabiYau dimension.
Triangulations of cyclic polytopes
, 2012
"... We give a new description of the combinatorics of triangulations of evendimensional cyclic polytopes, and of their bistellar flips. We show that the tropical exchange relation governing the number of intersections between diagonals of a polygon and a lamination (which generalizes to arbitrary surf ..."
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We give a new description of the combinatorics of triangulations of evendimensional cyclic polytopes, and of their bistellar flips. We show that the tropical exchange relation governing the number of intersections between diagonals of a polygon and a lamination (which generalizes to arbitrary surfaces) can also be generalized in a different way, to the setting of higher dimensional cyclic polytopes.
Thèse de doctorat Discipline: Mathématiques
"... Cette thèse concerne les algèbres amassées quantiques. Pour les algèbres amassées quantiques acycliques antisymétriques, nous exprimons les Fpolynômes quantiques et les monômes d’amas quantiques en termes des polynômes de Serre des grassmanniennes de carquois des modules rigides. Ensuite, nous intr ..."
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Cette thèse concerne les algèbres amassées quantiques. Pour les algèbres amassées quantiques acycliques antisymétriques, nous exprimons les Fpolynômes quantiques et les monômes d’amas quantiques en termes des polynômes de Serre des grassmanniennes de carquois des modules rigides. Ensuite, nous introduisons une nouvelle famille de variétés de carquois graduées avec une nouvelle tdéformation et généralisons les (q, t)caractères de Nakajima à ces constructions. Cela permet une approche par la (pseudo)catégorification monoidale déformée aux bases des algèbres amassées quantiques. Lorsque la graine initiale est acyclique, pour tout choix des coefficients et de la quantification, ces caractères nous donnent une base PBW duale, une base générique, et une base canonique duale avec des constantes de structure positives, les deux dernières bases contenant tous les monômes d’amas quantiques. Comme un sousproduit, nous obtenons la conjecture de positivité pour les algèbres amassées quantiques qui contiennent des graines acycliques. Motsclefs algèbre amassée quantique, variété de carquois, positivité, représentations de carquois, base canonique duale, caractères d’amas quantiques
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