Discrete Non-commutative Integrability: the Proof of a Conjecture by M. Kontsevich
نویسنده
چکیده
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.
منابع مشابه
Generalized Non-commutative Degeneration Conjecture
In this paper we propose a generalization of the Kontsevich–Soibelman conjecture on the degeneration of Hochschild-to-cyclic spectral sequence for smooth and compact DG category. Our conjecture states identical vanishing of a certain map between bi-additive invariants of arbitrary small DG categories over a field of characteristic zero. We show that this generalized conjecture follows from the ...
متن کاملA new proof for the Banach-Zarecki theorem: A light on integrability and continuity
To demonstrate more visibly the close relation between thecontinuity and integrability, a new proof for the Banach-Zareckitheorem is presented on the basis of the Radon-Nikodym theoremwhich emphasizes on measure-type properties of the Lebesgueintegral. The Banach-Zarecki theorem says that a real-valuedfunction $F$ is absolutely continuous on a finite closed intervalif and only if it is continuo...
متن کاملA short proof of Kontsevich’s cluster conjecture Une courte démonstration d’une conjoncture de Kontsevich
Article history: Received 18 November 2010 Accepted after revision 4 January 2011 Available online 20 January 2011 Presented by Maxime Kontsevich We give an elementary proof of the Kontsevich conjecture that asserts that the iterations of the noncommutative rational map Kr : (x, y) → (xyx−1, (1+ yr)x−1) are given by noncommutative Laurent polynomials. © 2011 Académie des sciences. Published by ...
متن کاملFORMALITY CONJECTURE by Maxim Kontsevich
This paper is devoted to a conjecture concerning the deformation quantization. This conjecture implies that arbitrary smooth Poisson manifold can be formally quantized, and the equivalence class of the resulting algebra is canonically defined. In other terms, it means that non-commutative geometry, in the formal approximation to the commutative geometry of smooth spaces, is described by the sem...
متن کاملA SHORT PROOF FOR THE EXISTENCE OF HAAR MEASURE ON COMMUTATIVE HYPERGROUPS
In this short note, we have given a short proof for the existence of the Haar measure on commutative locally compact hypergroups based on functional analysis methods by using Markov-Kakutani fixed point theorem.
متن کامل