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

Revising
Oct 11, 2014



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)

Incomplete paper
Nov, 2014


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. This paper predates the picture groups series of papers. But we are redoing the definition of propicture group to match the (finite type) picture groups definition.


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Picture groups of finite type and cohomology in type A_n. (with Kent Orr, Gordana Todorov, Jerzy Weyman)

IOTW4.v140925


For every quiver of finite type we define a finitely presented group called a picture group. We construct a finite CW complex which is shown in the signed exceptional sequences paper to be a K(pi,1) for this picture group. The special case of type A_n with straight orientation also has an independent proof in the noncrossing partitions paper. (But both of those other papers are based on ideas from a earlier version of this paper.) We use this cell complex to compute the cohomology of picture groups of type A_n with straight orientation. This paper needs to be updated following revision of its prequels [IT13], [IOTW3].


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

IT13.v150715 July 15, 2015


We introduce signed exceptional sequences as factorizations of morphisms in the cluster morphism category. The objects of this category are wide subcategories of the module category of a hereditary algebra. A morphism [T] : A → B is an equivalence class of rigid objects T in the cluster category of A so that B is the right homext perpendicular category of the underlying object T ∈ A. Factorizations of a morphism [T] are given by totally orderings of the components of T. This is equivalent to a “signed exceptional sequences.” For an algebra of finite representation type, the geometric realization of the cluster morphism category is the EilenbergMacLane space with fundamental group equal to the “picture group” introduced by the authors in [IOTW4]. The paper was updated following completion of IOTW3:1503.07945.


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Cluster morphism category as cubical category. (with Gordana Todorov)

Later


Using a result of Speyer and Thomas and Lemma 1.3.1 from ``Picture groups and maximal green sequences'' we show that, for modulated quivers of finite type, the classifying space of the cluster morphism category is a cubical category whose geometric realization is locally CAT(0). This gives another proof that the space X(Q) defined in the paper "Picture groups of finite type and cohomology in type A_n" is a K(pi,1).


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

IT14.v140923


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. In the new version, the statement of Lemma 1.3.1 is modified.

