Noncommutative Polygonal Cluster Algebras

TL;DR

This paper introduces polygonal cluster algebras, a noncommutative generalization of cluster algebras inspired by surface tilings, with applications to Lie groups and positive semigroups.

Contribution

It defines ST-compatible quivers represented by polygonal surface tilings, extending surface cluster algebra theory and connecting to Clifford algebras and $ heta$-positivity.

Findings

Construction of polygonal cluster algebras from surface tilings

Representation of these algebras in Clifford algebras

Development of noncommutative Somos sequences

Abstract

We define a new family of noncommutative generalizations of cluster algebras called polygonal cluster algebras. These algebras generalize the noncommutative surfaces of Berenstein-Retakh, and are inspired by the emerging theory of $\Theta$-positivity for the groups $\mathrm{Spin}(p,q)$. They are generated by mutations of quivers which we call ST-compatible, and which encode the order of the products that appear in the exchange relations. We show that these ST-compatible quivers can be represented by tilings of surfaces by polygons, a generalization of the description of surface type cluster algebras. As examples, we construct tilings which produce ST-compatible versions of the Del Pezzo quivers and the quivers first described by Le for Fock-Goncharov coordinates for Lie groups of type $B$. We show that polygonal cluster algebras have natural evaluations in Clifford algebras, which we use to produce noncommutative generalizations of the Somos sequences and to parameterize the $\Theta$-positive semigroup of $\mathrm{Spin}(2,n)$. We indicate how this will be done for the semigroup in $\mathrm{Spin}(p,q)$ and how one will give coordinates for general $\Theta$-positive representations into $\mathrm{Spin}(p,q)$.