Michael Shapiro (Michigan State University)
mshapiro@msu.edu
We define a discrete dynamical system (non commutative pentagram map) and prove its noncommutative integrability. To prove integrability we define non-commutative double quasi-Poisson bracket on the space of non-commutative arc weights of a directed graph on a cylinder which gives rise to the quasi Poisson bracket of Massuyeau and Turaev on the group algebra of the fundamental group of a surface. We show that the induced double quasi Poisson bracket on the boundary measurements can be described via non-commutative r-matrix formalism which gives a conceptual proof of the result by N. Ovenhouse that the traces of powers of Lax matrix form an infinity system of Hamiltonians in involution.
(Joint with N. Ovenhouse and S. Arthamonov.)