Trace pairings detecting Hochschild classes of zigzag algebras in every arity

Let Z be the zigzag algebra of a finite simple graph G with no isolated vertex, graded by path length and regarded as an A∞ algebra whose only operation is the product (TauCeti.zigzagStrictAInfinityAlgebra). A minimal A∞ deformation of Z whose first higher operation is m_{n+2} has that operation as its obstruction, a Hochschild cocycle in HH^{n+2, -n}(Z, Z). This file gives a supply of linear functionals on suspended Hochschild cochains which vanish on all coboundaries of degree zero, so that a cocycle on which one of them is nonzero is not a coboundary.

The word pairing

Fix weights a w on the words w = (w 0, …, w (n + 1)) of n + 2 darts of G. The word pairing (TauCeti.zigzagWordPairing) is

Λ_a(F) = ∑ w, a w · tr F(a_{w 0}, …, a_{w (n+1)}),

where a_d is the arrow of the dart d and tr is the Frobenius trace TauCeti.zigzagTrace, nonzero exactly on the volume classes. It vanishes on every Hochschild coboundary of degree zero (TauCeti.zigzagWordPairing_hochschildDifferential_eq_zero) as soon as the weights satisfy two conditions.

• Anticyclicity: moving the first dart of a word to the end changes the sign of its weight, a (u, e) = -a (e, u).
• The signless relation in every position: for every vertex v, the sum of the weights of the words carrying the backtrack d, d.symm based at v in two given consecutive positions, over all darts d with head v, vanishes.

On a word of arrows the Hochschild differential of a degree-zero suspended cochain φ is

δφ(x₀, …, x_{n+1}) = ∑_p φ(x₀, …, x_p x_{p+1}, …, x_{n+1})
                       - φ(x₀, …, x_n) x_{n+1} - x₀ φ(x₁, …, x_{n+1})

(TauCeti.zigzagStrictAInfinityAlgebra_hochschildDifferential_apply_of_tprod_of_mem). The two outer terms cancel in Λ_a by the trace property tr(xy) = tr(yx) and anticyclicity. In an inner term the product of two arrows is the volume class x_v when they form a backtrack based at v, and zero otherwise, so the inner terms cancel by the signless relation.

Weights from preprojective functionals

The second condition says that a vanishes on the signless preprojective relation ∑_{d.snd = v} d d.symm in every position, which is the local relation of the quadratic dual of Z found by Huerfano and Khovanov. Accordingly, let β send each dart to an element of a semiring B with a k-module structure, such as a k-algebra, such that ∑_{d.snd = v} β d * β d.symm = 0 at every vertex v: for example the classes of the arrows in the signless preprojective algebra TauCeti.signlessPreprojectiveAlgebra, or their matrices in one of its representations. Let f be a linear functional on B which is anticyclic on the arrows: f (y * β e) = -f (β e * y). Then the weights a w = f (β (w 0) ⋯ β (w (n + 1))) satisfy both conditions, and Λ_a vanishes on coboundaries (TauCeti.zigzagWordPairing_hochschildDifferential_eq_zero_of_anticyclic).

For instance, on a bipartite graph with a sign ε v = ±1 changing along every edge, the element s = ∑ ε v e_v of the signless preprojective algebra anticommutes with every arrow, so f y = g (s * y) is anticyclic on the arrows for every linear functional g vanishing on commutators. In this way cyclic classes of signless preprojective algebras in positive degree are paired with Hochschild classes of Z.

The backtrack pairing TauCeti.zigzagBacktrackPairing is the word pairing of length four with weights supported on backtrack walks.

Loop operations

On the loop deformation TauCeti.zigzagLoopAInfinityAlgebra with coefficients c on closed walks of length n + 2, the word pairing takes the value ± ∑ w, a w * c w (TauCeti.zigzagWordPairing_obstruction), the sign coming from the suspension. So the loop operation of a single closed walk w is not a coboundary as soon as f (β (w 0) ⋯ β (w (n + 1))) ≠ 0 for weights given by an anticyclic functional f.

Main definitions

• TauCeti.zigzagWordPairing: the word pairing Λ_a on suspended Hochschild cochains.

Main results

• TauCeti.zigzagWordPairing_hochschildDifferential_eq_zero: Λ_a vanishes on coboundaries of degree zero when its weights are anticyclic and satisfy the signless relation in every position.
• TauCeti.zigzagWordPairing_hochschildDifferential_eq_zero_of_anticyclic: the same for the weights given by an anticyclic functional on an algebra receiving the signless relations.
• TauCeti.not_exists_hochschildDifferential_eq_of_zigzagWordPairing_ne_zero: a cochain on which such a pairing is nonzero is not a Hochschild coboundary of degree zero.
• TauCeti.zigzagWordPairing_obstruction: the value of Λ_a on the higher operation of a loop deformation.
• not_exists_hochschildDifferential_eq_obstruction_zigzagLoopAInfinityAlgebra_of_anticyclic: a higher loop operation on which an anticyclic functional is nonzero is not a coboundary.

References

• R. S. Huerfano and M. Khovanov, A category for the adjoint representation, Section 4, for the quadratic dual of the zigzag algebra.
• Y. Liu and Z. Wang, A-infinity deformations of zigzag algebras via Ginzburg dg algebras, Sections 3 and 4, for the comparison of zigzag Hochschild cohomology with the cyclic quotient of the preprojective algebra.
• J.-L. Loday, Cyclic Homology, Chapter 1, for Hochschild chains and cochains and their duality.

The word pairing

noncomputable def TauCeti.zigzagWordPairing {k : Type w} [CommRing k] {V : Type u} {G : SimpleGraph V} [Finite V] (hns : ∀ (i : V), ∃ (j : V), G.Adj i j) [Fintype G.Dart] (n : ℕ) (a : (Fin (n + 2) → G.Dart) → k) : (ReducedTensorWords k (nonisolatedZigzagQuotient k G) →ₗ[k] nonisolatedZigzagQuotient k G) →ₗ[k] k

The word pairing with weights a w on the words of n + 2 darts: the linear functional on suspended Hochschild cochains F of the zigzag algebra

Λ_a(F) = ∑ w, a w · tr F(a_{w 0}, …, a_{w (n+1)}),

where a_d is the arrow of the dart d. It is the pairing, through the trace, with the Hochschild chain ∑ w, a w (1 ⊗ a_{w 0} ⊗ ⋯ ⊗ a_{w (n+1)}).

theorem TauCeti.zigzagWordPairing_apply {k : Type w} [CommRing k] {V : Type u} {G : SimpleGraph V} [Finite V] {hns : ∀ (i : V), ∃ (j : V), G.Adj i j} [Fintype G.Dart] (n : ℕ) (a : (Fin (n + 2) → G.Dart) → k) (F : ReducedTensorWords k (nonisolatedZigzagQuotient k G) →ₗ[k] nonisolatedZigzagQuotient k G) : (zigzagWordPairing hns n a) F = ∑ w : Fin (n + 2) → G.Dart, a w * (zigzagTrace k G hns) (F ((ReducedTensorWords.of k (nonisolatedZigzagQuotient k G) ⟨n + 2, ⋯⟩) ((PiTensorProduct.tprod k) fun (t : Fin ↑⟨n + 2, ⋯⟩) => (zigzagMk k G) (PathAlgebra.ofArrow (DoubledQuiver.arrow G ⋯)))))

Unfolding the word pairing.

theorem TauCeti.zigzagWordPairing_obstruction {k : Type w} [CommRing k] {V : Type u} {G : SimpleGraph V} [Finite V] {hns : ∀ (i : V), ∃ (j : V), G.Adj i j} [Fintype G.Dart] {n : ℕ} (hn : 1 ≤ n) (a : (Fin (n + 2) → G.Dart) → k) (c : (Fin (n + 2) → G.Dart) →₀ k) (hc : ∀ w ∈ c.support, ∀ (i : Fin (n + 2)), (w i).toProd.1 = (w (i + 1)).toProd.2) : (zigzagWordPairing hns n a) ((zigzagLoopAInfinityAlgebra hns c hc).obstruction (zigzagStrictAInfinityAlgebra k hns) ⟨n + 2, ⋯⟩) = ↑↑(∏ i : Fin (n + 2), (↑(n + 1 - ↑i)).negOnePow) * ∑ w : Fin (n + 2) → G.Dart, a w * c w

The word pairing of a higher loop operation. For n ≥ 1 and coefficients c on walks of length n + 2, the word pairing Λ_a of the operation of arity n + 2 of the loop deformation is ∑ w, a w * c w, up to the sign ∏ i, (-1) ^ (n + 1 - i) of the suspension.

theorem TauCeti.zigzagWordPairing_hochschildDifferential_eq_zero {k : Type w} [CommRing k] {V : Type u} {G : SimpleGraph V} [Finite V] {hns : ∀ (i : V), ∃ (j : V), G.Adj i j} [Fintype G.Dart] [DecidableEq V] {n : ℕ} {a : (Fin (n + 2) → G.Dart) → k} (hrot : ∀ (u : Fin (n + 1) → G.Dart) (e : G.Dart), a (Fin.snoc u e) = -a (Fin.cons e u)) (hrel : ∀ (p : Fin (n + 1)) (u : Fin n → G.Dart) (v : V), ∑ d : G.Dart with d.toProd.2 = v, a (p.castSucc.insertNth d (p.insertNth d.symm u)) = 0) {φ : ReducedTensorWords k (nonisolatedZigzagQuotient k G) →ₗ[k] nonisolatedZigzagQuotient k G} (hφ : LinearMap.IsHomogeneous φ (ReducedTensorWords.gradedPiece ((zigzagStrictAInfinityAlgebra k hns).grading.shift 1)) (((zigzagStrictAInfinityAlgebra k hns).grading.shift 1).piece ℤ) 0) : (zigzagWordPairing hns n a) ((zigzagStrictAInfinityAlgebra k hns).hochschildDifferential 0 φ) = 0

The word pairing vanishes on Hochschild coboundaries. Let the weights a on words of n + 2 darts be anticyclic, a (u, e) = -a (e, u), and satisfy the signless relation in every pair of consecutive positions: for every vertex v, the weights of the words carrying a backtrack d, d.symm based at v in the positions p and p + 1, summed over the darts d with head v, add up to zero. Then the word pairing Λ_a vanishes on the Hochschild differential of every suspended cochain of degree zero.

theorem TauCeti.not_exists_hochschildDifferential_eq_of_zigzagWordPairing_ne_zero {k : Type w} [CommRing k] {V : Type u} {G : SimpleGraph V} [Finite V] {hns : ∀ (i : V), ∃ (j : V), G.Adj i j} [Fintype G.Dart] [DecidableEq V] {n : ℕ} {a : (Fin (n + 2) → G.Dart) → k} (hrot : ∀ (u : Fin (n + 1) → G.Dart) (e : G.Dart), a (Fin.snoc u e) = -a (Fin.cons e u)) (hrel : ∀ (p : Fin (n + 1)) (u : Fin n → G.Dart) (v : V), ∑ d : G.Dart with d.toProd.2 = v, a (p.castSucc.insertNth d (p.insertNth d.symm u)) = 0) {F : ReducedTensorWords k (nonisolatedZigzagQuotient k G) →ₗ[k] nonisolatedZigzagQuotient k G} (hF : (zigzagWordPairing hns n a) F ≠ 0) : ¬∃ (φ : ReducedTensorWords k (nonisolatedZigzagQuotient k G) →ₗ[k] nonisolatedZigzagQuotient k G), LinearMap.IsHomogeneous φ (ReducedTensorWords.gradedPiece ((zigzagStrictAInfinityAlgebra k hns).grading.shift 1)) (((zigzagStrictAInfinityAlgebra k hns).grading.shift 1).piece ℤ) 0 ∧ (zigzagStrictAInfinityAlgebra k hns).hochschildDifferential 0 φ = F

A cochain detected by a word pairing is not a coboundary. If the weights a are anticyclic and satisfy the signless relation in every pair of consecutive positions, and the word pairing Λ_a does not vanish on F, then F is not the Hochschild differential of a suspended cochain of degree zero.

Weights from anticyclic functionals

theorem TauCeti.zigzagWordPairing_hochschildDifferential_eq_zero_of_anticyclic {k : Type w} [CommRing k] {V : Type u} {G : SimpleGraph V} [Finite V] {hns : ∀ (i : V), ∃ (j : V), G.Adj i j} {B : Type u_1} [Semiring B] [Module k B] [Fintype G.Dart] [DecidableEq V] {n : ℕ} (β : G.Dart → B) (hβ : ∀ (v : V), ∑ d : G.Dart with d.toProd.2 = v, β d * β d.symm = 0) (f : B →ₗ[k] k) (hf : ∀ (e : G.Dart) (y : B), f (y * β e) = -f (β e * y)) {φ : ReducedTensorWords k (nonisolatedZigzagQuotient k G) →ₗ[k] nonisolatedZigzagQuotient k G} (hφ : LinearMap.IsHomogeneous φ (ReducedTensorWords.gradedPiece ((zigzagStrictAInfinityAlgebra k hns).grading.shift 1)) (((zigzagStrictAInfinityAlgebra k hns).grading.shift 1).piece ℤ) 0) : (zigzagWordPairing hns n fun (w : Fin (n + 2) → G.Dart) => f (List.ofFn fun (t : Fin (n + 2)) => β (w t)).prod) ((zigzagStrictAInfinityAlgebra k hns).hochschildDifferential 0 φ) = 0

Weights from an anticyclic functional. Let β send each dart to an element of a semiring B with a k-module structure, such that the signless relation ∑_{d.snd = v} β d * β d.symm = 0 holds at every vertex v, and let f be a linear functional on B which is anticyclic on these elements: f (y * β e) = -f (β e * y). Then the word pairing with weights a w = f (β (w 0) ⋯ β (w (n + 1))) vanishes on the Hochschild differential of every suspended cochain of degree zero.

For the signless preprojective algebra of a bipartite graph and β the classes of the arrows, f can be a functional vanishing on commutators, twisted by the element ∑ ε v e_v for a sign ε changing along every edge.

theorem TauCeti.not_exists_hochschildDifferential_eq_obstruction_zigzagLoopAInfinityAlgebra_of_anticyclic {k : Type w} [CommRing k] {V : Type u} {G : SimpleGraph V} [Finite V] {hns : ∀ (i : V), ∃ (j : V), G.Adj i j} {B : Type u_1} [Semiring B] [Module k B] [Fintype G.Dart] [DecidableEq V] {n : ℕ} (hn : 1 ≤ n) (β : G.Dart → B) (hβ : ∀ (v : V), ∑ d : G.Dart with d.toProd.2 = v, β d * β d.symm = 0) (f : B →ₗ[k] k) (hf : ∀ (e : G.Dart) (y : B), f (y * β e) = -f (β e * y)) (c : (Fin (n + 2) → G.Dart) →₀ k) (hc : ∀ w ∈ c.support, ∀ (i : Fin (n + 2)), (w i).toProd.1 = (w (i + 1)).toProd.2) (h : ∑ w : Fin (n + 2) → G.Dart, f (List.ofFn fun (t : Fin (n + 2)) => β (w t)).prod * c w ≠ 0) : ¬∃ (φ : ReducedTensorWords k (nonisolatedZigzagQuotient k G) →ₗ[k] nonisolatedZigzagQuotient k G), LinearMap.IsHomogeneous φ (ReducedTensorWords.gradedPiece ((zigzagStrictAInfinityAlgebra k hns).grading.shift 1)) (((zigzagStrictAInfinityAlgebra k hns).grading.shift 1).piece ℤ) 0 ∧ (zigzagStrictAInfinityAlgebra k hns).hochschildDifferential 0 φ = (zigzagLoopAInfinityAlgebra hns c hc).obstruction (zigzagStrictAInfinityAlgebra k hns) ⟨n + 2, ⋯⟩

Higher loop operations detected by an anticyclic functional are not coboundaries. Let β satisfy the signless relations ∑_{d.snd = v} β d * β d.symm = 0 and let f be a linear functional anticyclic on the β e. For n ≥ 1 and coefficients c on closed walks of length n + 2, if ∑ w, f (β (w 0) ⋯ β (w (n + 1))) * c w ≠ 0, then the operation of arity n + 2 of the loop deformation with coefficients c, a Hochschild cocycle of the strict zigzag algebra by TauCeti.hochschildDifferential_obstruction_zigzagLoopAInfinityAlgebra, is not the Hochschild differential of any suspended cochain of degree zero.