Discrete non-commutative integrability: the proof of a conjecture by M. Kontsevich

We prove a conjecture of Kontsevich regarding the solutions of rank two recursion relations for non-commutative variables which, in the commutative case, reduce to rank two cluster algebras of affine type. The conjecture states that solutions are positive Laurent polynomials in the initial cluster v...

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Bibliographic Details
Main Authors Di Francesco, P, Kedem, R
Format Journal Article
LanguageEnglish
Published 03.09.2009
Subjects
Mathematics - Combinatorics
Mathematics - Mathematical Physics
Mathematics - Quantum Algebra
Physics - Mathematical Physics
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Summary:We prove a conjecture of Kontsevich regarding the solutions of rank two recursion relations for non-commutative variables which, in the commutative case, reduce to rank two cluster algebras of affine type. The conjecture states that solutions are positive Laurent polynomials in the initial cluster variables. We prove this by use of a non-commutative version of the path models which we used for the commutative case.
DOI:10.48550/arxiv.0909.0615