198 research outputs found
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
Contractible stability spaces and faithful braid group actions
We prove that any `finite-type' component of a stability space of a
triangulated category is contractible. The motivating example of such a
component is the stability space of the Calabi--Yau- category
associated to an ADE Dynkin quiver. In addition to
showing that this is contractible we prove that the braid group
acts freely upon it by spherical twists, in particular
that the spherical twist group is isomorphic to
. This generalises Brav-Thomas' result for the
case. Other classes of triangulated categories with finite-type components in
their stability spaces include locally-finite triangulated categories with
finite rank Grothendieck group and discrete derived categories of finite global
dimension.Comment: Final version, to appear in Geom. Topo
A survey on maximal green sequences
Maximal green sequences appear in the study of Fomin-Zelevinsky's cluster
algebras. They are useful for computing refined Donaldson-Thomas invariants,
constructing twist automorphisms and proving the existence of theta bases and
generic bases. We survey recent progress on their existence and properties and
give a representation-theoretic proof of Greg Muller's theorem stating that
full subquivers inherit maximal green sequences. In the appendix, Laurent
Demonet describes maximal chains of torsion classes in terms of bricks
generalizing a theorem by Igusa.Comment: 15 pages, submitted to the proceedings of the ICRA 18, Prague,
comments welcome; v2: misquotation in section 6 corrected; v3: minor changes,
final version; v4: reference to Jiarui Fei's work added, post-final version;
v4: formulation of Remark 4.3 corrected; v5: misquotation of Hermes-Igusa's
2019 paper corrected; v5: reference to Kim-Yamazaki's paper adde
Stability conditions and quantum dilogarithm identities for Dynkin quivers
We study fundamental group of the exchange graphs for the bounded derived
category D(Q) of a Dynkin quiver Q and the finite-dimensional derived category
D(\Gamma_N Q) of the Calabi-Yau-N Ginzburg algebra associated to Q. In the case
of D(Q), we prove that its space of stability conditions (in the sense of
Bridgeland) is simply connected; as applications, we show that its
Donanldson-Thomas invariant can be calculated via a quantum dilogarithm
function on exchange graphs. In the case of D(\Gamma_N Q), we show that
faithfulness of the Seidel-Thomas braid group action (which is known for Q of
type A or N = 2) implies the simply connectedness of its space of stability
conditions.Comment: Journal (almost) equivalent versio
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