Proof of a positivity conjecture of M. Kontsevich on non-commutative cluster variables

TL;DR

This paper proves Kontsevich's conjecture that iterations of a specific noncommutative rational map produce Laurent polynomials with nonnegative integer coefficients, confirming a positivity property in noncommutative cluster algebra theory.

Contribution

The paper establishes the positivity conjecture for noncommutative cluster variables associated with a particular rational map, advancing understanding in noncommutative algebra.

Findings

Iterations of the map are given by noncommutative Laurent polynomials.

Coefficients of these polynomials are nonnegative integers.

Confirmed the positivity conjecture of Kontsevich.

Abstract

We prove a conjecture of Kontsevich, which asserts that the iterations of the noncommutative rational map $F_r:(x,y)-->(xyx^{-1},(1+y^r)x^{-1})$ are given by noncommutative Laurent polynomials with nonnegative integer coefficients.