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 ‐positivity for the groups . 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 . 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 ‐positive semigroup of . We indicate how this will be done for the semigroup in and how one will give coordinates for general ‐positive representations into .