Exchange graphs and Ext quivers

By A. King, Yu Qiu. 2011

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Abstract We study the oriented exchange graph EG ∘ ( Γ N Q ) of reachable hearts in the finite-dimensional derived category D ( Γ N Q ) of the CY-N Ginzburg algebra Γ N Q associated to an acyclic quiver Q. We show that any such heart is induced from some heart in the bounded derived category D ( Q ) via some ‘Lagrangian immersion’ L : D ( Q ) → D ( Γ N Q ) . We build on this to show that the quotient of EG ∘ ( Γ N Q ) by the Seidel–Thomas braid group is the exchange graph CEG N − 1 ( Q ) of cluster tilting sets in the (higher) cluster category C N − 1 ( Q ) . As an application, we interpret Buan–Thomas' coloured quiver for a cluster tilting set in terms of the Ext quiver of any corresponding heart in D ( Γ N Q ) .

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/ca2fcababc4e3517f622e7d42612b39fe9e98b06