Bipartiteness and the signless preprojective relation

The local preprojective relator of a finite quiver Q at a vertex v carries a sign,

ρ_v = ∑_{head a = v} a a* - ∑_{tail a = v} a* a,

whereas the signless relator s_v = ∑_{head a = v} a a* + ∑_{tail a = v} a* a of the doubled quiver Quiver.Symmetrify Q — the relation which appears in the quadratic dual of a zigzag algebra — carries none. Rescaling the arrows of Q by scalars ε replaces ρ by the gauged relator ρ_ε, and for a bipartite Q the colour signs make the corner of ρ_ε at every vertex a multiple of s_v, by TauCeti.gaugedPreprojectiveRelator_bipartite_vertexCorner_eq_smul.

This file proves the obstruction supplied by an odd closed walk. The backtracks at a fixed vertex are distinct basis paths, so the corner equation e_v ρ_ε e_v = c_v • s_v reads off the gauge: ε_a is the scalar c at the head of a, and -c at its tail. Hence c changes sign along every arrow and is a unit at each vertex incident to an arrow as soon as the corresponding value of ε is a unit. A closed walk of odd length in the doubled quiver therefore forces 2 = 0 in the coefficient ring. A loop is the smallest such walk.

Main results

• TauCeti.gauge_eq_of_vertexCorner_eq_smul: a cornerwise comparison with the signless relator reads off the gauge at every arrow meeting that vertex.
• TauCeti.eq_neg_of_forall_vertexCorner_eq_smul: the comparison scalars change sign along every arrow.
• TauCeti.not_exists_forall_vertexCorner_eq_smul_of_odd_length: no unit gauge compares the preprojective relation cornerwise with the signless one when the doubled quiver carries a closed walk of odd length, unless 2 = 0.

References

S. Huerfano and M. Khovanov, A category for the adjoint representation, Section 3, https://arxiv.org/abs/math/0002060, for the signless relation of the quadratic dual of a zigzag algebra and its comparison with the preprojective relation of a bipartite graph. The preprojective conventions follow Crawley-Boevey, Quiver algebras, weighted projective lines, and the Deligne--Simpson problem, Section 1.

Reading off the gauge

theorem TauCeti.gauge_eq_of_vertexCorner_eq_smul (k : Type w) {Q : Type u} [CommRing k] [Quiver Q] [Fintype Q] [(i j : Q) → Fintype (i ⟶ j)] (ε : ⦃i j : Q⦄ → (i ⟶ j) → k) {v : Q} {c : k} (hv : doubledVertexIdempotent k v * gaugedPreprojectiveRelator k ε * doubledVertexIdempotent k v = c • signlessPreprojectiveRelator k (Quiver.Symmetrify.of.obj v)) : (∀ (i : Q) (a : i ⟶ v), ε a = c) ∧ ∀ (j : Q) (a : v ⟶ j), ε a = -c

A cornerwise comparison of the gauged relator with the signless relator reads off the gauge. If the corner of ρ_ε at v is the multiple c • s_v of the signless relator, then ε is c on every arrow into v and -c on every arrow out of v.

theorem TauCeti.eq_neg_of_forall_vertexCorner_eq_smul (k : Type w) {Q : Type u} [CommRing k] [Quiver Q] [Fintype Q] [(i j : Q) → Fintype (i ⟶ j)] (ε : ⦃i j : Q⦄ → (i ⟶ j) → k) {c : Q → k} (hc : ∀ (w : Q), doubledVertexIdempotent k w * gaugedPreprojectiveRelator k ε * doubledVertexIdempotent k w = c w • signlessPreprojectiveRelator k (Quiver.Symmetrify.of.obj w)) {i j : Q} (a : i ⟶ j) : c j = -c i

The comparison scalars change sign along every arrow. If the corner of the gauged relator at every vertex is the corresponding multiple of the signless relator, then the scalars c of that comparison satisfy c j = -c i for every arrow a : i ⟶ j.

theorem TauCeti.isUnit_of_forall_vertexCorner_eq_smul (k : Type w) {Q : Type u} [CommRing k] [Quiver Q] [Fintype Q] [(i j : Q) → Fintype (i ⟶ j)] {ε : ⦃i j : Q⦄ → (i ⟶ j) → k} {i j : Q} (a : i ⟶ j) (hεa : IsUnit (ε a)) {c : Q → k} (hc : ∀ (w : Q), doubledVertexIdempotent k w * gaugedPreprojectiveRelator k ε * doubledVertexIdempotent k w = c w • signlessPreprojectiveRelator k (Quiver.Symmetrify.of.obj w)) : IsUnit (c j)

The comparison scalar at the head of an arrow is the gauge of that arrow, hence a unit whenever the gauge is.

theorem TauCeti.not_exists_forall_vertexCorner_eq_smul_of_odd_length (k : Type w) {Q : Type u} [CommRing k] [Quiver Q] [Fintype Q] [(i j : Q) → Fintype (i ⟶ j)] {ε : ⦃i j : Q⦄ → (i ⟶ j) → k} (hε : ∀ ⦃i j : Q⦄ (a : i ⟶ j), IsUnit (ε a)) (h2 : 2 ≠ 0) {v : Q} (p : Quiver.Path (Quiver.Symmetrify.of.obj v) (Quiver.Symmetrify.of.obj v)) (hp : Odd p.length) : ¬∃ (c : Q → k), ∀ (w : Q), doubledVertexIdempotent k w * gaugedPreprojectiveRelator k ε * doubledVertexIdempotent k w = c w • signlessPreprojectiveRelator k (Quiver.Symmetrify.of.obj w)

A closed walk of odd length obstructs the signless comparison. If the doubled quiver has a closed walk of odd length at some vertex, then over a coefficient ring in which 2 ≠ 0 no unit gauge ε makes the corner of ρ_ε at every vertex a multiple of the signless relator there. This complements the positive bipartite comparison TauCeti.gaugedPreprojectiveRelator_bipartite_vertexCorner_eq_smul.

theorem TauCeti.not_exists_forall_vertexCorner_eq_smul_of_loop (k : Type w) {Q : Type u} [CommRing k] [Quiver Q] [Fintype Q] [(i j : Q) → Fintype (i ⟶ j)] {ε : ⦃i j : Q⦄ → (i ⟶ j) → k} (hε : ∀ ⦃i j : Q⦄ (a : i ⟶ j), IsUnit (ε a)) (h2 : 2 ≠ 0) {v : Q} (a : v ⟶ v) : ¬∃ (c : Q → k), ∀ (w : Q), doubledVertexIdempotent k w * gaugedPreprojectiveRelator k ε * doubledVertexIdempotent k w = c w • signlessPreprojectiveRelator k (Quiver.Symmetrify.of.obj w)

A quiver with a loop admits no signless comparison. A loop is a closed walk of length one in the doubled quiver, so it obstructs the comparison over every coefficient ring in which 2 ≠ 0.