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Discrete non-commutative integrability: proof of a conjecture by M. Kontsevich. (English) Zbl 1276.16025
Summary: We prove a conjecture of Kontsevich regarding the solutions of rank 2 recursion relations for non-commutative variables, which, in the commutative case, reduce to rank 2 cluster algebras of affine type. The conjecture states that solutions are positive Laurent polynomials in the initial cluster variables. We prove this by the use of a non-commutative version of the path models, which we used for the commutative case.

MSC:
16S38 Rings arising from noncommutative algebraic geometry
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
13F60 Cluster algebras
05C90 Applications of graph theory
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