The famous differential Painlevé equations define the most general non-linear special functions. They are connected to various branches of mathematics as well as mathematical physics. In recent years, there has been growing interest in generalizing these equations to the non-commutative case, motivated by problematics and needs of modern physics. Furthermore, it is natural to study the properties of these non-commutative analogs, which extend those in the commutative setting. In particular, one may inquire about the non-commutative analogs of the so-called monodromy manifolds associated with the isomonodromic problems of the Painlevé equations.
In this talk, we will discuss a method for deriving these analogs and defining the associated Poisson structure. The talk is based on the paper arXiv:2302.10694 and an ongoing project in collaboration with S. Arthamonov.
