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

Part of: Algebraic combinatorics Rings and algebras arising under various constructions Arithmetic rings and other special rings

Published online by Cambridge University Press:  31 October 2012

Kyungyong Lee  and
Ralf Schiffler
Kyungyong Lee
Affiliation:
Department of Mathematics, Wayne State University, Detroit, MI 48202, USA (email: klee@math.wayne.edu)
Ralf Schiffler
Affiliation:
Department of Mathematics, University of Connecticut, Storrs, CT 06269, USA (email: schiffler@math.uconn.edu)
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Abstract

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We prove a conjecture of Kontsevich, which asserts that the iterations of the non-commutative rational map Fr:(x,y)→(xyx−1,(1+yr)x−1) are given by non-commutative Laurent polynomials with non-negative integer coefficients.

Keywords

positivitynon-commutative Laurent polynomiallattice path

MSC classification

Secondary: 16S38: Rings arising from non-commutative algebraic geometry 05E99: None of the above, but in this section 13F60: Cluster algebras
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 6 , November 2012 , pp. 1821 - 1832
Copyright
Copyright © Foundation Compositio Mathematica 2012

References

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