Non-commutative integrability, paths and quasi-determinants

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In previous work, we showed that the solution of certain systems of discrete integrable equations, notably Q and T-systems, is given in terms of partition functions of positively weighted paths, thereby proving the positive Laurent phenomenon of Fomin and Zelevinsky for these cases. This method of solution is amenable to generalization to non-commutative weighted paths. Under certain circumstances, these describe solutions of discrete evolution equations in non-commutative variables: Examples are the corresponding quantum cluster algebras (Berenstein and Zelevinsky (2005) [3]), the Kontsevich evolution (Di Francesco and Kedem (2010) [10]) and the T-systems themselves (Di Francesco and Kedem (2009) [8]). In this paper, we formulate certain non-commutative integrable evolutions by considering paths with non-commutative weights, together with an evolution of the weights that reduces to cluster algebra mutations in the commutative limit. The general weights are expressed as Laurent monomials of quasi-determinants of path partition functions, allowing for a non-commutative version of the positive Laurent phenomenon. We apply this construction to the known systems, and obtain Laurent positivity results for their solutions in terms of initial data.

Original languageEnglish (US)
Pages (from-to)97-152
Number of pages56
JournalAdvances in Mathematics
Issue number1
StatePublished - Sep 10 2011


  • Cluster algebra
  • Continued fractions
  • Discrete integrable systems
  • Noncommutative
  • Path models
  • Quantum cluster algebra
  • Quasi-determinants

ASJC Scopus subject areas

  • Mathematics(all)


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