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Continuous cluster categories II: Continuous clustertilted categories. (with Gordana Todorov)

later



Since CCCI has now appeared, we are working to finish part II. This paper starts with a review of the definitions and results of CCC I. This revised version has complete proofs of the main theorems, namely (1) the rational and continuous clustertilted categories are abelian and (2) they are isomorphic to categories of string modules over the Jacobian algebra of an infinite quiver with potential. Further revisions are needed to incorporate the precise descritption of the triangulation of the continuous categories given by the continuous Frobenius category.


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Periodic trees and propictures. (with Kent Orr, Gordana Todorov, Jerzy Weyman)

later


This is a continuation of the paper Periodic trees and semiinvariants. The semiinvariant picture for an affine quiver is a inverse limit of pictures for an inverse system of finitely presented groups. The walls of the compartments separating the periodic functions for each period tree are separated by domains of these semiinvariants.


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Cohomology of finite picture groups. (with Kent Orr, Gordana Todorov, Jerzy Weyman)

later


Every quiver of finite type has a finitely presented group called a picture group. We construct a finite CW complex which is a K(pi,1) for this picture group and use it to compute the cohomology of picture groups of type A_n. Although the group depends on the orientation of the quiver, its cohomology depends only on n.


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Signed exceptional sequences and the cluster morphism category. (with Gordana Todorov)

later


To every modulated quiver we associate the category whose objects are the finitely generated wide subcategories of the module category and whose morphisms M1 > M2 are the partial cluster tilting objects of M1 whose right perpendicular category is M2. A sequence of composable morphisms is thus a signed exceptional sequence, the composition is the corresponding (unordered) cluster. (There is a bijection between signed exceptional sequences and ordered clusters.)


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The category of noncrossing partitions.

CAT0.v140911


This category has noncrossing partitions of n+1 as objects and binary forests as morphisms. I use a theorem of Gromov to show that the classifying space of this category is locally CAT(0). It is not too hard to show that the fundamental group of this space is the picture group of type An with straight orientation. In an appendix (not ready), jointly written with Gordana Todorov, we use representation theory and a theorem of Speyer and Thomas to extend this to picture groups of finite type. They are all CAT(0) groups.


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Picture groups and maximal green sequences. (with Gordana Todorov)

IT14.v140910


It is easy to see that there is a monomorphism from the set of maximal green sequences for an acyclic quiver into the set of positive expressions for the coxeter element of the corresponding picture group. This paper show that, in finite type, this mapping is a bijection.

