Higher Hochschild classes of zigzag algebras from matrix representations

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). This file turns a matrix representation of the signless preprojective algebra of a bipartite subgraph of G into Hochschild classes of Z: a closed walk whose matrix product has nonzero trace has a loop operation which is not a Hochschild coboundary.

Let f be a copy of a bipartite graph H in G, not necessarily induced, and let β x y be matrices on the arrows of H satisfying the signless relations ∑_{y ∼ x} β y x * β x y = 0. Extend β by zero to G (SimpleGraph.Copy.extendByZero) and send each dart d of G to the block matrix E_{d.snd, d.fst} ⊗ β d.fst d.snd, indexed by pairs of a vertex of G and a matrix index. These block matrices satisfy the signless relations of G, which are the local relations of the quadratic dual of Z. Twist the trace by the diagonal matrix D with the sign ±1 of the bipartition of H along the copy, and 1 elsewhere:

F(Y) = tr(D Y).

Moving the block matrix of a dart d across D changes the sign ε d.fst into ε d.snd, which is its negative whenever the block is nonzero, since then d is the image of an edge of H. So F is anticyclic on the block matrices, and its word pairing (TauCeti.zigzagWordPairing_hochschildDifferential_eq_zero_of_anticyclic) vanishes on Hochschild coboundaries. On the loop operation of the image of a closed walk w of H, of length n + 2, it takes the value ±tr(β (w 0) ⋯ β (w (n + 1))), in Tau Ceti's later-factor-first order.

This is the Hochschild counterpart of TauCeti.exists_mem_signlessPreprojectiveGrade_notMem_commutatorSubmodule_of_trace_ne_zero, which reads the same trace as a nonzero class in the cyclic quotient of the signless preprojective algebra. Liu and Wang identify that cyclic quotient with the second Hochschild cohomology of the two-dimensional Ginzburg algebra, and through derived Koszul duality with the Hochschild classes of Z which obstruct its intrinsic formality.

Main results

• not_exists_hochschildDifferential_eq_obstruction_zigzagLoopAInfinityAlgebra_of_trace_ne_zero: the loop operation of the image of a closed walk of a bipartite subgraph, whose matrix product in a representation of the signless relations has nonzero trace, is not a Hochschild coboundary.
• not_exists_hochschildDifferential_eq_obstruction_zigzagLoopAInfinityAlgebra_of_trace_eq_one: the same for integer matrices whose product has trace one, over every nontrivial commutative ring.

References

• Y. Liu and Z. Wang, A-infinity deformations of zigzag algebras via Ginzburg dg algebras, https://arxiv.org/abs/2301.06757, Sections 3 and 4, for the comparison of zigzag Hochschild cohomology with the cyclic quotient of the preprojective algebra.
• R. S. Huerfano and M. Khovanov, A category for the adjoint representation, Section 4, for the quadratic dual of the zigzag algebra.

Block matrices of darts

Non-coboundaries from closed walks

theorem TauCeti.not_exists_hochschildDifferential_eq_obstruction_zigzagLoopAInfinityAlgebra_of_trace_ne_zero {k : Type w} [CommRing k] {V : Type u} {G : SimpleGraph V} {W : Type u_1} [Fintype W] {H : SimpleGraph W} [DecidableRel H.Adj] {m : Type u_2} [Fintype m] [Finite V] {hns : ∀ (i : V), ∃ (j : V), G.Adj i j} [DecidableEq m] (f : H.Copy G) (hH : H.IsBipartite) (β : W → W → Matrix m m k) (hβ : ∀ (x : W), ∑ y ∈ H.neighborFinset x, β y x * β x y = 0) {n : ℕ} (hn : 1 ≤ n) {w : Fin (n + 2) → H.Dart} (hw : ∀ (i : Fin (n + 2)), (w i).toProd.1 = (w (i + 1)).toProd.2) (htr : (List.ofFn fun (t : Fin (n + 2)) => β (w t).toProd.1 (w t).toProd.2).prod.trace ≠ 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 (Finsupp.single (fun (t : Fin (n + 2)) => f.toHom.mapDart (w t)) 1) ⋯).obstruction (zigzagStrictAInfinityAlgebra k hns) ⟨n + 2, ⋯⟩

Higher Hochschild classes of a zigzag algebra from a matrix representation of a bipartite subgraph. Let f be a copy of a bipartite graph H in G, not necessarily induced, and let β be a family of matrices on H satisfying the signless relations ∑_{y ∼ x} β y x * β x y = 0. Let w be a closed walk of H of length n + 2, with n ≥ 1, read in product order, such that β (w 0) ⋯ β (w (n + 1)) has nonzero trace. Then the loop operation of the image of w in G, the operation of arity n + 2 of the loop deformation with coefficient one on that walk, is a Hochschild cocycle of the strict zigzag algebra of G (TauCeti.hochschildDifferential_obstruction_zigzagLoopAInfinityAlgebra) which is not the Hochschild differential of any suspended cochain of degree zero.

theorem TauCeti.not_exists_hochschildDifferential_eq_obstruction_zigzagLoopAInfinityAlgebra_of_trace_eq_one {k : Type w} [CommRing k] {V : Type u} {G : SimpleGraph V} {W : Type u_1} [Fintype W] {H : SimpleGraph W} {m : Type u_2} [Fintype m] [Finite V] {hns : ∀ (i : V), ∃ (j : V), G.Adj i j} [DecidableEq m] [Nontrivial k] (f : H.Copy G) (hH : H.IsBipartite) (β : W → W → Matrix m m ℤ) (h₀ : ∀ (x y : W), ¬H.Adj x y → β x y = 0) (hβ : ∀ (x : W), ∑ y : W, β y x * β x y = 0) {n : ℕ} (hn : 1 ≤ n) {w : Fin (n + 2) → H.Dart} (hw : ∀ (i : Fin (n + 2)), (w i).toProd.1 = (w (i + 1)).toProd.2) (htr : (List.ofFn fun (t : Fin (n + 2)) => β (w t).toProd.1 (w t).toProd.2).prod.trace = 1) : ¬∃ (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 (Finsupp.single (fun (t : Fin (n + 2)) => f.toHom.mapDart (w t)) 1) ⋯).obstruction (zigzagStrictAInfinityAlgebra k hns) ⟨n + 2, ⋯⟩

Higher Hochschild classes from an integer matrix representation of a bipartite subgraph. An integer matrix family on a bipartite graph H which vanishes off the edges and satisfies the signless relations, together with a closed walk of H of length n + 2, with n ≥ 1, whose matrix product has trace one, gives along any copy of H in G a loop operation of the strict zigzag algebra of G which is not a Hochschild coboundary, over every nontrivial commutative ring.

theorem TauCeti.not_hochschildVanishesAt_zigzagStrictAInfinityAlgebra_of_trace_ne_zero {k : Type w} [CommRing k] {V : Type u} {G : SimpleGraph V} {W : Type u_1} [Fintype W] {H : SimpleGraph W} [DecidableRel H.Adj] {m : Type u_2} [Fintype m] [Finite V] {hns : ∀ (i : V), ∃ (j : V), G.Adj i j} [DecidableEq m] (β : W → W → Matrix m m k) (hβ : ∀ (x : W), ∑ y ∈ H.neighborFinset x, β y x * β x y = 0) (f : H.Copy G) (hH : H.IsBipartite) {n : ℕ} (hn : 1 ≤ n) {u : Fin (n + 2) → H.Dart} (hu : ∀ (i : Fin (n + 2)), (u i).toProd.1 = (u (i + 1)).toProd.2) (htr : (List.ofFn fun (t : Fin (n + 2)) => β (u t).toProd.1 (u t).toProd.2).prod.trace ≠ 0) : ¬(zigzagStrictAInfinityAlgebra k hns).HochschildVanishesAt ⟨n + 2, ⋯⟩

A detected loop gives a nonzero higher Hochschild class.

theorem TauCeti.not_hochschildVanishesAt_zigzagStrictAInfinityAlgebra_of_trace_eq_one {k : Type w} [CommRing k] {V : Type u} {G : SimpleGraph V} {W : Type u_1} [Fintype W] {H : SimpleGraph W} {m : Type u_2} [Fintype m] [Finite V] {hns : ∀ (i : V), ∃ (j : V), G.Adj i j} [DecidableEq m] [Nontrivial k] (β : W → W → Matrix m m ℤ) (h₀ : ∀ (x y : W), ¬H.Adj x y → β x y = 0) (h : ∀ (x : W), ∑ y : W, β y x * β x y = 0) (f : H.Copy G) (hH : H.IsBipartite) {n : ℕ} (hn : 1 ≤ n) {u : Fin (n + 2) → H.Dart} (hu : ∀ (i : Fin (n + 2)), (u i).toProd.1 = (u (i + 1)).toProd.2) (htr : (List.ofFn fun (t : Fin (n + 2)) => β (u t).toProd.1 (u t).toProd.2).prod.trace = 1) : ¬(zigzagStrictAInfinityAlgebra k hns).HochschildVanishesAt ⟨n + 2, ⋯⟩

A detected loop gives a nonzero higher Hochschild class.

theorem TauCeti.not_hochschildVanishesAt_four_zigzagStrictAInfinityAlgebra_of_isContained {k : Type w} [CommRing k] {V : Type u} {G : SimpleGraph V} [Finite V] {hns : ∀ (i : V), ∃ (j : V), G.Adj i j} [Nontrivial k] {n : ℕ} (hn : 4 ≤ n) (hG : (AffineDynkinType.D n).graph.IsContained G) : ¬(zigzagStrictAInfinityAlgebra k hns).HochschildVanishesAt ⟨4, not_hochschildVanishesAt_four_zigzagStrictAInfinityAlgebra_of_isContained._proof_1⟩

Graphs containing an affine Dₙ have HH^{4, -2} nonzero. If a graph G with no isolated vertex contains a copy of the affine diagram Dₙ (n ≥ 4), not necessarily induced, then over every nontrivial commutative ring the Hochschild cohomology HH^{4, -2} of the strict zigzag algebra of G is nonzero: the loop operation of the closed walk 1 → 0 → 1 → (n - 1) → 1 at the image of the branch node 1 is a Hochschild cocycle which is not a coboundary.