Networks and Noncommutative Integrable Systems

Given an edge-weighted graph drawn on a torus (or a cylinder), one gets an associated Liouville integrable system. These are known as "cluster integrable systems" because they are preserved by local moves on the graph which correspond to cluster algebra mutations. Goncharov and Kenyon studied these systems in relation with the dimer model, interpreting the Hamiltonians in terms of perfect matchings, and Gekhtman-Shapiro-Vainshtein described them in terms of directed paths. We will study a noncommutative version of these cluster integrable systems, where the edge weights on the graph are formal noncommutative variables. In joint work with Shapiro and Arthamonov, we study a certain non-commutative Poisson structure on this space, and give results analogous to the commutative case, including an R-matrix formula for the brackets, and a set of commuting Hamiltonians which can be interpreted in terms of paths.