Noncommutative Networks on a Cylinder

In this paper a double quasi Poisson bracket in the sense of Van den Bergh is constructed on the space of noncommutative weights of arcs of a directed graph embedded in a disk or cylinder \(\Sigma \), which gives rise to the quasi Poisson bracket of G. Massuyeau and V. Turaev on the group algebra \(\textbf{k}\pi _1(\Sigma ,p)\) of the fundamental group of a surface based at \(p\in \partial \Sigma \). This bracket also induces a noncommutative Goldman Poisson bracket on the cyclic space \(\mathcal C_\natural \), which is a \({\textbf{k}}\)-linear space of unbased loops. We show that the induced double quasi Poisson bracket between boundary measurements can be described via noncommutative r-matrix formalism. This gives a more conceptual proof of the result of Ovenhouse (Adv Math 373:107309, 2020) that traces of powers of Lax operator form an infinite collection of noncommutative Hamiltonians in involution with respect to noncommutative Goldman bracket on \(\mathcal C_\natural \).