23 research outputs found
Cluster algebras of infinite rank as colimits
We formalize the way in which one can think about cluster algebras of
infinite rank by showing that every rooted cluster algebra of infinite rank can
be written as a colimit of rooted cluster algebras of finite rank. Relying on
the proof of the posivity conjecture for skew-symmetric cluster algebras (of
finite rank) by Lee and Schiffler, it follows as a direct consequence that the
positivity conjecture holds for cluster algebras of infinite rank. Furthermore,
we give a sufficient and necessary condition for a ring homomorphism between
cluster algebras to give rise to a rooted cluster morphism without
specializations. Assem, Dupont and Schiffler proposed the problem of a
classification of ideal rooted cluster morphisms. We provide a partial solution
by showing that every rooted cluster morphism without specializations is ideal,
but in general rooted cluster morphisms are not ideal.Comment: Included cluster algebras of uncountable rank, fixed some typos.
Results on the countable case unchanged, comments appreciate
Mutation of torsion pairs in cluster categories of Dynkin type
Mutation of torsion pairs in triangulated categories and its combinatorial
interpretation for the cluster category of Dynkin type and of type
have been studied by Zhou and Zhu. In this paper we present a
combinatorial model for mutation of torsion pairs in the cluster category of
Dynkin type , using Ptolemy diagrams of Dynkin type which were
introduced by Holm, J{\o}rgensen and Rubey.Comment: Corrected typos, some arguments made more concise, results unchange
Homotopy invariants of singularity categories
We present a method for computing -homotopy invariants of
singularity categories of rings admitting suitable gradings. Using this we
describe any such invariant, e.g. homotopy K-theory, for the stable categories
of self-injective algebras admitting a connected grading. A remark is also made
concerning the vanishing of all such invariants for cluster categories of type
quivers.Comment: final revisio
Cluster tilting modules for mesh algebras
We study cluster tilting modules in mesh algebras of Dynkin type, providing a
new proof for their existence. In all but one case, we show that these are
precisely the maximal rigid modules, and that they are equivariant for a
certain automorphism. We further study their mutation, providing an example of
mutation in an abelian category which is not stably 2-Calabi-Yau, and
explicitly describe the combinatorics.Comment: comments appreciated; the third version includes a discussion on the
combinatorics of the mutation
Lattices of t-structures and thick subcategories for discrete cluster categories
We classify t-structures and thick subcategories in discrete cluster
categories of Dynkin type , and show that the set
of all t-structures on is a lattice under inclusion
of aisles, with meet given by their intersection. We show that both the lattice
of t-structures on obtained in this way and the
lattice of thick subcategories of are intimately
related to the lattice of non-crossing partitions of type . In particular,
the lattice of equivalence classes of non-degenerate t-structures on such a
category is isomorphic to the lattice of non-crossing partitions of a finite
linearly ordered set.Comment: 21 pages, comments welcom
On the graded dual numbers, arcs, and non-crossing partitions of the integers
We give a combinatorial model for the bounded derived category of graded modules over the dual numbers in terms of arcs on the integer line with a point at infinity. Using this model we describe the lattice of thick subcategories of the bounded derived category, and of the perfect complexes, in terms of non-crossing partitions. We also make some comments on the symmetries of these lattices, exceptional collections, and the analogous problem for the unbounded derived category
Approximating triangulated categories by spaces
We initiate a systematic study of lattices of thick subcategories for
arbitrary essentially small triangulated categories. To this end we give
several examples illustrating the various properties these lattices may, or may
not, have and show that as soon as a lattice of thick subcategories is
distributive it is automatically a spatial frame. We then construct two
non-commutative spectra, one functorial and one with restricted functoriality,
that give universal approximations of these lattices by spaces.Comment: Further minor updates, accepted for publication in Advances in
Mathematic