On Cluster -Algebras

We introduce a -algebra attached to the cluster and a quiver . If is the quiver coming from triangulation of the Riemann surface with a finite number of cusps, we prove that the primitive spectrum of times is homeomorphic to a generic subset of the Teichmüller space of surface . We conclude with an analog of the Tomita-Takesaki theory and the Connes invariant for the algebra .

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