Noncommutative surfaces, clusters, and their symmetries

Abstract: The aim of my talk (based on joint work in progress with Min Huang and Vladimir Retakh) is to introduce and study certain noncommutative algebras $A$ for any marked surface. These algebras admit noncommutative clusters, i.e., embeddings of a given group $G$ which is either free or one-relator (we call it triangle group) into the multiplicative monoid $A^\times$. The clusters are parametrized by triangulations of the surface and exhibit a noncommutative Laurent Phenomenon, which asserts that generators of the algebra can be written as sums of the images of elements of $G$ for any noncommutative cluster. If the surface is unpunctured, then our algebra $A$ can be specialized to the ordinary quantum cluster algebra, and the noncommutative Laurent Phenomenon becomes the (positive) quantum one.

It turns out that there is a natural action of a certain braid-like group $Br_A$ by automorphisms of $G$ on each cluster in a compatible way (this is, indeed, the braid group $Br_n$ if the surface is an unpunctured disk with n+2 marked boundary points). If surface is punctured, the algebra $A$ admits a family of commuting automorphisms which will give new clusters and new "tagged" noncommutative Laurent Phenomena.

There are important elements in $A$ assigned to each marked point, which we refer to as noncommutative angles (or h-lengths). They belong to the group algebra of each cluster group and are invariant under all noncommutative cluster mutations. This eventually gives rise to noncommutative integrable systems on unpunctured cylinders and other surfaces which, in particular, recover the ones introduced by Kontsevich in 2011 together with their Laurentness and positivity.

mathematical physics

Audience: researchers in the topic

Seminar-Type Workshop on Noncommutative Integrable Systems