Noncommutative integrability, paths and quasi-determinants

Abstract

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 [BZ], the Kontsevich evolution [DFK09b] and the $T$-systems themselves [DFK09a]. 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.

46 pages, minor typos corrected

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• Non-commutative Hermite--Padé approximation and integrability
• Quantum Q systems: From cluster algebras to quantum current algebras
• Proof of a positivity conjecture of M. Kontsevich on non-commutative cluster variables
• Hopf algebra structure of generalized quasi-symmetric functions in partially commutative variables
• The solution of the quantum $A_1$ T-system for arbitrary boundary
• Solutions to the ultradiscrete Toda molecule equation expressed as minimum weight flows of planar graphs
• Quantum Q-systems and fermionic sums -- the non-simply laced case
• Inverse moment problem for non-Abelian Coxeter double Bruhat cells
• Matrix Orthogonal Polynomials, non-abelian Toda lattice and Bäcklund transformation