Ordered Exchange Graphs

By Thomas Brustle, Dong Yang. 2013

118 citations recorded by the source corpus.

The exchange graph of a cluster algebra encodes the combinatorics of mutations of clusters. Through the recent "categorifications" of cluster algebras using representation theory one obtains a whole variety of exchange graphs associated with objects such as a finite-dimensional algebra or a differential graded algebra concentrated in non-positive degrees. These constructions often come from variations of the concept of tilting, the vertices of the exchange graph being torsion pairs, t-structures, silting objects, support $\tau$-tilting modules and so on. All these exchange graphs stemming from representation theory have the additional feature that they are the Hasse quiver of a partial order which is naturally defined for the objects. In this sense, the exchange graphs studied in this article can be considered as a generalization or as a completion of the poset of tilting modules which has been studied by Happel and Unger. The goal of this article is to axiomatize the thus obtained structure of an ordered exchange graph, to present the various constructions of ordered exchange graphs and to relate them among each other.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9192c6b4179deda1c090c02366dfcfe93fc50644

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Citations in the corpus

The wall-chamber structures of the real Grothendieck groups (2019, methodology): “The answer is given by the following result of [BY].”
Exchange graphs and Ext quivers (2011, background): “rem 6.7) is to prove that Q+(J(Hb)) = ΠN(Qε(Hb)), and so deduce that Qε(H) = Q+(α(H)), providing the promised interpretation of the coloured quiver. The CY-3 case is much more studied (as surveyed in [4]) and is in many ways more uniform. Indeed, we go on to show (Theorem 9.5) that EG◦ 3(Γ Q,H) is a”
Reduction of τ-Tilting Modules and Torsion Pairs (2013, background): “We refer the reader to [9] for an in-depth survey of the relations between these objects and several other important concepts in representation theory.”
Lattice Properties of Oriented Exchange Graphs and Torsion Classes (2015, methodology): “The next result, which appears in [4] in much greater generality, shows that oriented exchange graphs can be studied using functorially finite torsion classes of Λ.”
Silting objects, simple-minded collections, $t$-structures and co-$t$-structures for finite-dimensional algebras (2012, background): “. They are also closely related to fundamental concepts in cluster theory such as clusters ([20]), c-matrices and g-matrices ([21, 40]) and cluster-tilting objects ([7]). We refer to the survey paper [16] for more details. A concrete example to be given at the end of the article demonstrates one pra”
Classifying tilting complexes over preprojective algebras of Dynkin type (2015, background): “Furthermore, silting objects have turned out to have deep connections with several important objects such as cluster tilting objects and t-structures, for example [Adachi et al. 2014; Buan et al. 2011; Koenig and Yang 2014; Brüstle and Yang 2013; Iyama et al. 2014; Qiu and Woolf 2014; Broomhead et a”
Silting Objects (2018, background): “For further details we refer to [96] and to the survey [38].”