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1	Cluster structures for 2-Calabi-Yau categories and unipotent groups Scott, J, Reiten, I, Iyama, O, Buan, A.B. 12 1900 (has links) No description available. AR-quivers cluster tilting cluster categories cluster algebras quiver mutation reduced expressions 2-Calabi–Yau categories preprojective algebras
2	Exchange Graphs via Quiver Mutation Warkentin, Matthias 02 October 2014 (has links) (PDF) Inspired by Happel's question, whether the exchange graph and the simplicial complex of tilting modules over a quiver algebra are independent from the multiplicities of multiple arrows in the quiver, we study quantitative aspects of Fomin and Zelevinsky's quiver mutation rule. Our results turn out to be very useful in the mutation-infinite case for understanding combinatorial structures as the cluster exchange graph or the simplicial complex of tilting modules, which are governed by quiver mutation. Using a class of quivers we call forks we can show that any such quiver yields a tree in the exchange graph. This allows us to provide a good global description of the exchange graphs of arbitrary mutation-infinite quivers. In particular we show that the exchange graph of an acyclic quiver is a tree if (and in fact only if) any two vertices are connected by at least two arrows. Furthermore we give classification results for the simplicial complexes and thereby obtain a partial positive answer to Happel's question. Another consequence of our findings is a confirmation of Unger's conjecture about the infinite number of components of the tilting exchange graph in all but finitely many cases. Finally we generalise and conceptualise our results by introducing what we call "polynomial quivers", stating several conjectures about "polynomial quiver mutation", and giving proofs in special cases. Köchermutation Austauschgraph Baum Cluster-Algebra Cluster-Komplex Kippmodul simplizialer Komplex Polynomköcher Astgabel mutations-unendlich Mehrfachpfeil quiver mutation exchange graph tree cluster algebra cluster complex tilting module simplicial complex polynomial quiver fork mutation-infinite multiple arrow ddc:512 ddc:514 Kombinatorik Darstellungstheorie Simplizialer Komplex Graph Baum
3	Exchange Graphs via Quiver Mutation Warkentin, Matthias 11 June 2014 (has links) Inspired by Happel's question, whether the exchange graph and the simplicial complex of tilting modules over a quiver algebra are independent from the multiplicities of multiple arrows in the quiver, we study quantitative aspects of Fomin and Zelevinsky's quiver mutation rule. Our results turn out to be very useful in the mutation-infinite case for understanding combinatorial structures as the cluster exchange graph or the simplicial complex of tilting modules, which are governed by quiver mutation. Using a class of quivers we call forks we can show that any such quiver yields a tree in the exchange graph. This allows us to provide a good global description of the exchange graphs of arbitrary mutation-infinite quivers. In particular we show that the exchange graph of an acyclic quiver is a tree if (and in fact only if) any two vertices are connected by at least two arrows. Furthermore we give classification results for the simplicial complexes and thereby obtain a partial positive answer to Happel's question. Another consequence of our findings is a confirmation of Unger's conjecture about the infinite number of components of the tilting exchange graph in all but finitely many cases. Finally we generalise and conceptualise our results by introducing what we call "polynomial quivers", stating several conjectures about "polynomial quiver mutation", and giving proofs in special cases. info:eu-repo/classification/ddc/512 ddc:512 info:eu-repo/classification/ddc/514 ddc:514

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