Noncommutative Polygonal Cluster Algebras

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 $Θ$-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 $Θ$-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 $Θ$-positive representations into $\mathrm{Spin}(p,q)$.

Figures (46)

• Figure 1: A decorated polygon and associated ST-compatible quiver.
• Figure 2: Fan triangulation of a disk with 5 marked points on the boundary and associated quiver.
• Figure 3: Flips are related to exchange relations.
• Figure 4: Quivers associated to the cluster structure for decorated representations into split real forms of (A) $B_2$ and (B) $B_3$.
• Figure 5: Mutation at 3 "mixes" layers of quiver.

Theorems & Definitions (243)

• Theorem : \ref{['thm:seedalgebra_iso']}
• Theorem : \ref{['thm:laurent']}
• Example
• Theorem : \ref{['prop:so2n_warmpu']}
• Remark 1.1
• Remark 2.1
• Definition 2.2
• Proposition 2.3
• Proposition 2.4
• Definition 2.5