Species with Potential Arising from Surfaces with Orbifold Points of Order 2, Part II: Arbitrary Weights

Abstract

Let Σ = (Σ, M, O) be either an unpunctured surface with marked points and order-2 orbifold points, or a once-punctured closed surface with order-2 orbifold points. For each pair (τ, ω) consisting of a triangulation τ of Σ and a function ω : O → {1, 4}, we define a chain complex C•(τ, ω) with coefficients in F2 = Z/2Z. Given Σ and ω, we define a colored triangulation of Σω = (Σ, M, O, ω) to be a pair (τ, ξ) consisting of a triangulation of Σ and a 1-cocycle in the cochain complex which is dual to C•(τ, ω); the combinatorial notion of colored flip of colored triangulations is then defined as a refinement of the notion of flip of triangulations. Our main construction associates to each colored triangulation a species and a potential, and our main result shows that colored triangulations related by a flip have species with potentials (SPs) related by the corresponding SP-mutation as defined in [27].We define the flip graph of colored triangulations of Σω as the graph whose vertices are the pairs (τ, x) consisting of a triangulation τ and a cohomology class x ∈ H 1 (C • (τ, ω)), with an edge connecting two such pairs (τ, x) and (σ, z) if and only if there exist 1-cocycles ξ ∈ x and ζ ∈ z such that (τ, ξ) and (σ, ζ) are colored triangulations related by a colored flip; then we prove that this flip graph is always disconnected provided the underlying surface Σ is not contractible.In the absence of punctures, we show that the Jacobian algebras of the SPs constructed are finite-dimensional and that whenever two colored triangulations have the same underlying triangulation, the Jacobian algebras of their associated SPs are isomorphic if and only if the underlying 1-cocycles have the same cohomology class; we also give a full classification of the non-degenerate SPs one can associate to any given pair (τ, ω) over cyclic Galois extensions with primitive 4 th roots of unity.The species constructed here are species realizations of the 2 |O| skew-symmetrizable matrices that Felikson-Shapiro-Tumarkin associated in [18] to any given triangulation of Σ. In the prequel [27] to this paper we constructed a species realization of only one of these matrices, but therein we allowed the presence of arbitrarily many punctures.

Species with potential arising from surfaces with orbifold points of order 2, part I: one choice of weights

“…In this paper we will restrict our attention to only one of these matrices, namely, the matrix B(τ ) corresponding to the weighted quiver (Q(τ ), d(τ )) defined above. In the forthcoming [20] we will consider (the weighted quiver of) other skew-symmetrizable matrices. (4) Let us describe more precisely how Felikson-Shapiro-Tumarkin associate several skew-symmetrizable matrices to each tagged triangulation τ .…”

“…In this paper we have considered only one of these matrices. In the forthcoming sequel [20] to this paper we will present a construction of SPs for triangulations that will encompass other matrices. In particular, we will give constructions of SPs for the skew-symmetrizable matrices that are mutation-equivalent to matrices of types B and B.…”

Semilinear clannish algebras arising from surfaces with orbifold points

“…In [19,Definition 3.5] and [20], the tensor ring R A(Q, d, g) (resp. the complete tensor ring R A(Q, d, g) ) is called the path algebra (resp.…”

“…With (Q(τ, ω), d(τ, ω)) at hand, in Subsection 6.2 we associate a Jacobian algebra ( §6.2.1) and a semilinear clannish algebra ( §6.2.2) to each colored triangulation. More specifically, in §6.2.1 we recall from [20] how for each 1-cocycle ξ of C • (τ ) one can associate a species with potential (A(τ, ξ), W (τ, ξ)) to the colored triangulation (τ, ξ). This is done by taking a degree-d field extension E/F from Section 3, where d := lcm{d(τ, ω) k | k ∈ τ }, and then attaching to each k ∈ τ the unique subfield F k of E such that [F k : F ] = d(τ, ω) k , and to each arrow a : k → j of Q(τ ) the F j -F k -bimodule A(τ, ξ) a := F g(τ,ξ)a j ⊗ Fj ∩F k F k , where g(τ, ξ) a ∈ Gal(F j ∩ F k /F ) is either an extension of θ ξa : L → L to…”

“…The species with potential associated to (τ, ξ) in §6.2.1 (resp. §6.3.1) was first constructed in [20] (resp. [19]).…”