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Non-commutative integrability, paths and quasi-determinants. (English) Zbl 1252.37069
Summary: 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 [A. Berenstein and A. Zelevinsky, Adv. Math. 195, No. 2, 405–455 (2005; Zbl 1124.20028)], the Kontsevich evolution [P. Di Francesco and R. Kedem, Int. Math. Res. Not. 2010, No. 21, 4042–4063 (2010; Zbl 1276.16025)] and the \(T\)-systems themselves [P. Di Francesco and R. Kedem, Electron. J. Comb. 16, No. 1, Research Paper R140, 39 p. (2009: Zbl 1229.13019)]. 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.

MSC:
37K60 Lattice dynamics; integrable lattice equations
13F60 Cluster algebras
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