Results 1 - 10
of
10
Cotorsion pairs in the cluster category of a marked surface
- Department of Mathematics, University of Leicester, University Road, Leicester
"... Dedicated to Professor Idun Reiten on the occasion of her seventieth birthday. We study extension spaces, cotorsion pairs and their mutations in the cluster category of a marked surface without punctures. Under the one-to-one correspondence between the curves, valued closed curves in the marked surf ..."
Abstract
-
Cited by 4 (3 self)
- Add to MetaCart
Dedicated to Professor Idun Reiten on the occasion of her seventieth birthday. We study extension spaces, cotorsion pairs and their mutations in the cluster category of a marked surface without punctures. Under the one-to-one correspondence between the curves, valued closed curves in the marked surface and the indecomposable objects in the associated cluster category, we prove that the dimension of extension space of two indecom-posable objects in the cluster categories equals to the intersection number of the correspond-ing curves. By using this result, we prove that there are no non-trivial t−structures in the cluster categories when the surface is connected. Based on this result, we give a classifi-cation of cotorsion pairs in these categories. Moreover we define the notion of paintings of a marked surface without punctures and their rotations. They are a geometric model of cotorsion pairs and of their mutations respectively.
Cluster algebras arising from cluster tubes
, 2014
"... We study the cluster algebras arising from cluster tubes with rank bigger than 1. Cluster tubes are 2-Calabi–Yau triangulated categories that contain no cluster tilting objects, but maximal rigid objects. Fix a certain maximal rigid object T in the cluster tube Cn of rank n. For any indecomposable ..."
Abstract
- Add to MetaCart
We study the cluster algebras arising from cluster tubes with rank bigger than 1. Cluster tubes are 2-Calabi–Yau triangulated categories that contain no cluster tilting objects, but maximal rigid objects. Fix a certain maximal rigid object T in the cluster tube Cn of rank n. For any indecomposable rigid objectM in Cn, we define an analogousXM of Caldero–Chapoton’s formula (or Palu’s cluster character formula) by using the geometric information of M. We show that XM, XM ′ satisfy the mutation formula when M,M ′ form an exchange pair, and that X?:M → XM gives a bijection from the set of indecomposable rigid objects in Cn to the set of cluster variables of cluster algebra of type Cn−1, which induces a bijection between the set of basic maximal rigid objects in Cn and the set of clusters. This yields a surprising result proved recently by Buan–Marsh–Vatne that the combinatorics of maximal rigid objects in the cluster tube Cn encodes the combinatorics of the cluster algebra of type Bn−1, since the combinatorics of cluster algebras of type Bn−1 and of type Cn−1 is the same by a result of Fomin and Zelevinsky. As a consequence, we give a categorification of cluster algebras of type C.
Thèse de doctorat Discipline: Mathématiques
"... Cette thèse concerne les algèbres amassées quantiques. Pour les algèbres amassées quantiques acycliques antisymétriques, nous exprimons les F-polynômes quantiques et les monômes d’amas quantiques en termes des polynômes de Serre des grassmanniennes de carquois des modules rigides. Ensuite, nous intr ..."
Abstract
- Add to MetaCart
(Show Context)
Cette thèse concerne les algèbres amassées quantiques. Pour les algèbres amassées quantiques acycliques antisymétriques, nous exprimons les F-polynômes 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 t-déformation 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 sous-produit, nous obtenons la conjecture de positivité pour les algèbres amassées quantiques qui contiennent des graines acycliques. Mots-clefs algèbre amassée quantique, variété de carquois, positivité, représentations de carquois, base canonique duale, caractères d’amas quantiques
