Quivers with potentials associated to triangulated surfaces, part IV: removing boundary assumptions

Abstract

We prove that the quivers with potentials associated with triangulations of surfaces with marked points, and possibly empty boundary, are non-degenerate, provided the underlying surface with marked points is not a closed sphere with exactly five punctures. This is done by explicitly defining the QPs that correspond to tagged triangulations and proving that whenever two tagged triangulations are related to a flip, their associated QPs are related to the corresponding QP-mutation. As a byproduct, for (arbitrarily punctured) surfaces with non-empty boundary, we obtain a proof of the non-degeneracy of the associated QPs which is independent from the one given by the author in the first paper of the series. The main tool used to prove the aforementioned compatibility between flips and QP-mutations is what we have called Popping Theorem, which, roughly speaking, says that an apparent lack of symmetry in the potentials arising from ideal triangulations with self-folded triangles can be fixed by a suitable right-equivalence.

Tagged mapping class groups: Auslander–Reiten translation

“…In [22] it is further shown that the corresponding quivers with potential are also related by a sequence of mutations, and it follows then from [17] that the corresponding categories D f d (Γ) are equivalent as triangulated categories, likewise for C (Γ). We therefore write simply D(S) = D f d (Γ) and C (S) = C (Γ) in this case.…”

The representation type of Jacobian algebras

“…Now choose any puncture q. A minor modification of the proof of [LF3,Proposition 11.2] proves that S(τ, x) is right equivalent to S(τ, w), where w = (w p ) p∈P is defined by w q = p∈P x p , and w p = 1 for p = q.…”

“…Given the definition of the QPs associated to tagged triangulations (cf. [LF3,Definition 3.2]), this implies that both S(τ, x) and S(σ, x) are non-degenerate (recall that τ and σ have been assumed to be tagged triangulations).…”

“…However, [LF2] does not deal with the case of flips involving arbitrary tagged triangulations of the sphere with five punctures. This case is not dealt with either in the more recent paper [LF3].…”

On Jacobian algebras associated with the once-punctured torus

“…Still, in particular its Jacobian algebras are not well understood, yet. Consider the two non-degenerate potentials W := γ 1 β 1 α 1 + γ 2 β 2 α 2 W ′ := γ 1 β 1 α 1 + γ 2 β 2 α 2 − γ 2 β 1 α 2 γ 1 β 2 α 1 yielding one infinite dimensional Jacobian algebra Λ = P(Q, W ) [DWZ1,Example 8.6] and one finite-dimensional Jacobian algebra Λ ′ = P(Q, W ′ ) [La,Example 8.2]. Recently Geuenich [G] proved that there are infinitely many non-degenerate potentials for Q up to right equivalence.…”