Non-commutative Networks on a Cylinder

La exposición es parte del Online Cluster Algebra Seminar, en cuyo comité organizador está involucrado Daniel Labardini.

Non-commutative Networks on a Cylinder

This is a joint work with S. Artamоnov and N. Ovenhouse. We constructed a double quasi Poisson bracket in the sense of Van den Bergh on the space of non-commutative weights of arcs of a directed graph embedded in a disk or cylinder Σ. This bracket gives rise to the quasi Poisson bracket of G. Massuyeau and V.Turaev on the group algebra kπ1(Σ,p) of the fundamental group of a surface based at p∈∂Σ and induces a non-commutative Goldman Poisson bracket on the cyclic space C♮, which is a k-linear space of unbased loops. We show that the induced double quasi Poisson bracket between boundary measurements can be described via non-commutative r-matrix formalism.

This gives a more conceptual proof of N. Ovenhouse's result that traces of powers of Lax matrix for pentagram system form an infinite collection of non-commutative Hamiltonians in involution with respect to non-commutative Goldman bracket on C♮.