Noncommutative integrability, paths and quasi-determinants

TL;DR

This paper extends the path-based solutions of integrable systems to non-commutative variables, demonstrating a non-commutative Laurent positivity phenomenon and connecting to quantum cluster algebras.

Contribution

It introduces a framework for non-commutative integrable evolutions using quasi-determinants, generalizing cluster algebra mutations and positive Laurent phenomena.

Findings

Established non-commutative path solutions for integrable systems

Proved Laurent positivity in non-commutative setting

Connected non-commutative solutions to quantum cluster algebras

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.