Catégorification d'algèbres amassées antisymétrisables

Abstract : The aim of this thesis is to categorify some skew-symmetrizable cluster algebras. A lot of skew-symmetric cases have been handled for example by Keller, Caldero-Keller, Geiß-Leclerc-Schröer, Dehy-Keller, Fu-Keller, Palu. In order to do that, one uses stably 2-Calabi-Yau exact categories. For the skew-symmetrizable case, we consider an action of a finite group on such a category and we introduce an equivariant category which is also stably 2-Calabi-Yau. We develop a theory of mutations for its invariant rigid objects. A large familly of examples is given by categories of representations of preprojective algebras : for instance, the category of representations of the preprojective algebra of type A(2n-1) with its automorphism of order 2 gives rise to the cluster algebra of functions over the unipotent Lie group of type C(n). In a similar way, we can get all the cluster algebras of functions over unipotent maximal subgroups of semi-simple Lie groups. Moreover, we obtain all the cluster algebras of finite type. All these categorifications lead to a proof, for the corresponding cluster algebras, of a conjecture of Fomin and Zelevinsky which states that the cluster monomials are linearly independant.
Document type :
Mathématiques [math]. Université de Caen, 2008. Français
Contributor : Laurent Demonet <>
Submitted on : Tuesday, January 6, 2009 - 9:21:27 AM
Last modification on : Tuesday, January 6, 2009 - 9:28:24 AM
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  • HAL Id : tel-00350137, version 1



Laurent Demonet. Catégorification d'algèbres amassées antisymétrisables. Mathématiques [math]. Université de Caen, 2008. Français. <tel-00350137>



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