The Broken Ptolemy Algebra: a Finite-type Laurent Phenomenon Algebra

Type A, or Ptolemy cluster algebras are a prototypical example of finite type cluster algebras, as introduced by Fomin and Zelevinsky. Their combinatorics is that of triangulations of a polygon. Lam and Pylyavskyy have introduced a generalization of cluster algebras where the exchange polynomials are not necessarily binomial, called Laurent phenomenon algebras. It is an interesting and hard question to classify finite type Laurent phenomenon algebras. Here we show that “breaking” one of the arrows in a type A mutation class quiver surprisingly yields a finite type LP algebra, whose combinatorics can still be understood in terms of diagonals and triangulations of a polygon.