A∞ deformations of a zigzag algebra supported on closed walks

Let Z be the zigzag algebra of a finite simple graph G with no isolated vertex, graded by path length and regarded as a cohomologically graded algebra: vertex idempotents in degree 0, arrows in degree 1 and volume classes in degree 2. This is the regrading of an arrow to bidegree (1, -1), read in the cohomological component.

A loop operation is attached to n ≥ 1 darts w 0, …, w (n - 1) of G. It is the n-linear map

m_w(x₀, …, x_{n-1}) = a_{w 0}(x₀) ⋯ a_{w (n - 1)}(x_{n-1}) · x_v,

where a_d is the coordinate of the arrow of d in the vertex-arrow-volume basis and x_v is the volume class at the head v of w 0. On arrows it is nonzero only on the word w itself. When the darts form a closed walk, read in Tau Ceti's later-factor-first order (w (n - 1) is traversed first, and (w i).fst = (w (i + 1)).snd cyclically), the operation is a Hochschild cocycle (TauCeti.zigzagLoopOperation_cocycle). Its coboundary telescopes, because the arrow coordinate of a product only sees the idempotents at the two ends of the arrow (TauCeti.zigzagBasis_coord_dart_mul). Moreover m_w ∘ m_w vanishes, since m_w takes values in the volume classes, which have no arrow coordinates.

Consequently, for n ≥ 3, every finite linear combination μ of loop operations of closed walks of length n gives an A∞ algebra TauCeti.zigzagLoopAInfinityAlgebra on Z. Its operations are m₁ = 0, m₂ the product, mₙ = μ, and m r = 0 otherwise. It is minimal and strictly unital with unit 1, so it is a strictly unital minimal A∞ deformation of the strict zigzag algebra. Its operation m r has cohomological degree 2 - r, and degree r - 2 for the Adams grading, which is the negative of path length. For walks of length one or two the coefficients are not used: there are no closed walks of length one in a simple graph, and the loop operation of a closed walk of length two, a backtrack, would change the product rather than add a higher operation.

Such deformations include the quartic operation of Liu--Wang's Example 4.7 for affine D₄; see TauCeti.RepresentationTheory.Quiver.Zigzag.AInfinity.Affine.D.Four. Longer closed walks give higher operations: for instance, closed walks of lengths 6, 8 and 12 match the path degrees of the nonzero cyclic classes of the preprojective algebras of the affine E₆, E₇ and E₈ graphs.

Some of the quartic deformations are trivial modulo A∞ isomorphism (TauCeti.RepresentationTheory.Quiver.Zigzag.AInfinity.Backtrack) and, when 2 is invertible, some are not (TauCeti.RepresentationTheory.Quiver.Zigzag.AInfinity.NonFormal).

Main definitions

• TauCeti.zigzagLoopOperation: the loop operation of n darts.
• TauCeti.zigzagLoopAInfinityAlgebra: the A∞ algebra with m₂ the product and mₙ a linear combination of loop operations of closed walks of length n.
• TauCeti.zigzagStrictAInfinityAlgebra: the strict zigzag algebra, the loop deformation with zero coefficients.

Main results

• TauCeti.zigzagLoopOperation_ofArrow: a loop operation is nonzero on arrows only on its own word, where it is the volume class at the base of the walk.
• TauCeti.zigzagLoopOperation_cocycle: the loop operation of a closed walk is a Hochschild cocycle.
• TauCeti.zigzagLoopAInfinityAlgebra_m_two_apply, TauCeti.zigzagLoopAInfinityAlgebra_m_eq_linearCombination and TauCeti.zigzagLoopAInfinityAlgebra_m_eq_zero: the operations.
• TauCeti.zigzagLoopAInfinityAlgebra_m_mem: the higher operation takes values in the volume classes.
• TauCeti.zigzagLoopAInfinityAlgebra_grading_eq: the grading depends neither on the coefficients nor on the length of the walks.
• TauCeti.zigzagStrictAInfinityAlgebra_m_two_apply and TauCeti.zigzagStrictAInfinityAlgebra_m_eq_zero: the strict zigzag algebra has no operation besides the product m₂.
• TauCeti.isMinimal_zigzagLoopAInfinityAlgebra and TauCeti.strictUnit_one_zigzagLoopAInfinityAlgebra: the algebra is minimal and strictly unital.
• TauCeti.isHomogeneous_zigzagLoopAInfinityAlgebra_m_adams: the Adams degree of m r is r - 2.

References

• Y. Liu and Z. Wang, A-infinity deformations of zigzag algebras via Ginzburg dg algebras, Section 4, for minimal A∞ deformations of zigzag algebras and Example 4.7.
• B. Keller, Introduction to A-infinity algebras and modules, Section 3.1, for the Stasheff identities.

Loop operations

noncomputable def TauCeti.zigzagLoopOperation (k : Type w) [CommRing k] {V : Type u} (G : SimpleGraph V) [Finite V] (hns : ∀ (i : V), ∃ (j : V), G.Adj i j) {n : ℕ} [NeZero n] (w : Fin n → G.Dart) : (Fin n → nonisolatedZigzagQuotient k G) →ₘₗ[k] nonisolatedZigzagQuotient k G

The loop operation of n darts w: the product of the w i-arrow coordinates of the n inputs, times the volume class at the head of w 0.

@[simp]

theorem TauCeti.zigzagLoopOperation_apply (k : Type w) [CommRing k] {V : Type u} (G : SimpleGraph V) [Finite V] (hns : ∀ (i : V), ∃ (j : V), G.Adj i j) {n : ℕ} [NeZero n] (w : Fin n → G.Dart) (x : Fin n → nonisolatedZigzagQuotient k G) : (zigzagLoopOperation k G hns w) x = (∏ i : Fin n, ((zigzagBasis k G hns).coord (Sum.inr (Sum.inl (w i)))) (x i)) • zigzagVolume k G (w 0).toProd.2

A loop operation is the product of the arrow coordinates of its inputs, times the volume class at the head of w 0.

theorem TauCeti.zigzagLoopOperation_ofArrow (k : Type w) [CommRing k] {V : Type u} (G : SimpleGraph V) [Finite V] (hns : ∀ (i : V), ∃ (j : V), G.Adj i j) {n : ℕ} [NeZero n] (w d : Fin n → G.Dart) : ((zigzagLoopOperation k G hns w) fun (i : Fin n) => (zigzagMk k G) (PathAlgebra.ofArrow (DoubledQuiver.arrow G ⋯))) = if d = w then zigzagVolume k G (w 0).toProd.2 else 0

A loop operation on arrows: on the arrows of n darts d, the loop operation of w is the volume class at the head of w 0 when d = w, and zero otherwise.

The cocycle identity

theorem TauCeti.zigzagLoopOperation_cocycle (k : Type w) [CommRing k] {V : Type u} (G : SimpleGraph V) [Finite V] (hns : ∀ (i : V), ∃ (j : V), G.Adj i j) {n : ℕ} [NeZero n] {w : Fin n → G.Dart} (hw : ∀ (i : Fin n), (w i).toProd.1 = (w (i + 1)).toProd.2) (x : ℕ → nonisolatedZigzagQuotient k G) : (∑ p ∈ Finset.range n, negOnePowCast k ↑p • (zigzagLoopOperation k G hns w).evalNat (replaceBlock x p 2 (x p * x (p + 1))) + negOnePowCast k ↑n • ((zigzagLoopOperation k G hns w).evalNat x * x n) - x 0 * (zigzagLoopOperation k G hns w).evalNat fun (j : ℕ) => x (1 + j)) = 0

The loop operation of a closed walk is a Hochschild cocycle. For a closed walk w of length n and inputs x 0, …, x n, with L = m_w,

∑_{p < n} (-1)^p L(x₀, …, x_p x_{p+1}, …, x_n) + (-1)^n L(x₀, …, x_{n-1}) x_n - x₀ L(x₁, …, x_n)

vanishes. The p-th term is the sum of the terms in which the vertex idempotent at the p-th or (p + 1)-st vertex of the walk is inserted, so the alternating sum telescopes.

The operations of the deformation

Degrees

The Stasheff identities

The A∞ algebra

noncomputable def TauCeti.zigzagLoopAInfinityAlgebra {k : Type w} [CommRing k] {V : Type u} {G : SimpleGraph V} [Finite V] (hns : ∀ (i : V), ∃ (j : V), G.Adj i j) {n : ℕ} [NeZero n] (c : (Fin n → G.Dart) →₀ k) (hc : ∀ w ∈ c.support, ∀ (i : Fin n), (w i).toProd.1 = (w (i + 1)).toProd.2) : AInfinityAlgebra k (nonisolatedZigzagQuotient k G)

The loop deformation of a zigzag algebra. For coefficients c supported on closed walks of length n ≥ 3, read in product order, this is the A∞ algebra on the zigzag quotient of G. It has the path-length grading, m₂ the product, mₙ the linear combination of the loop operations TauCeti.zigzagLoopOperation with coefficients c, and every other operation zero. For n ≤ 2 the coefficients are not used, and this is the strict zigzag algebra.

theorem TauCeti.fst_eq_snd_of_mem_support_single {k : Type w} [CommRing k] {V : Type u} {G : SimpleGraph V} {n : ℕ} [NeZero n] {w : Fin n → G.Dart} (hw : ∀ (i : Fin n), (w i).toProd.1 = (w (i + 1)).toProd.2) (r : k) (w' : Fin n → G.Dart) : w' ∈ (Finsupp.single w r).support → ∀ (i : Fin n), (w' i).toProd.1 = (w' (i + 1)).toProd.2

A single coefficient on a closed walk is supported on closed walks, so it defines a loop deformation.

@[simp]

theorem TauCeti.zigzagLoopAInfinityAlgebra_grading_piece {k : Type w} [CommRing k] {V : Type u} {G : SimpleGraph V} [Finite V] {hns : ∀ (i : V), ∃ (j : V), G.Adj i j} {n : ℕ} [NeZero n] (c : (Fin n → G.Dart) →₀ k) (hc : ∀ w ∈ c.support, ∀ (i : Fin n), (w i).toProd.1 = (w (i + 1)).toProd.2) : (zigzagLoopAInfinityAlgebra hns c hc).grading.piece ℤ = zigzagIntegerGrade k G

The loop deformation is graded by path length.

theorem TauCeti.zigzagLoopAInfinityAlgebra_grading_eq {k : Type w} [CommRing k] {V : Type u} {G : SimpleGraph V} [Finite V] {hns : ∀ (i : V), ∃ (j : V), G.Adj i j} {n : ℕ} [NeZero n] (c : (Fin n → G.Dart) →₀ k) (hc : ∀ w ∈ c.support, ∀ (i : Fin n), (w i).toProd.1 = (w (i + 1)).toProd.2) {n' : ℕ} [NeZero n'] (c' : (Fin n' → G.Dart) →₀ k) (hc' : ∀ w ∈ c'.support, ∀ (i : Fin n'), (w i).toProd.1 = (w (i + 1)).toProd.2) : (zigzagLoopAInfinityAlgebra hns c hc).grading = (zigzagLoopAInfinityAlgebra hns c' hc').grading

The grading of a loop deformation depends neither on its coefficients nor on the length of its walks.

@[simp]

theorem TauCeti.zigzagLoopAInfinityAlgebra_m_two_apply {k : Type w} [CommRing k] {V : Type u} {G : SimpleGraph V} [Finite V] {hns : ∀ (i : V), ∃ (j : V), G.Adj i j} {n : ℕ} [NeZero n] (c : (Fin n → G.Dart) →₀ k) (hc : ∀ w ∈ c.support, ∀ (i : Fin n), (w i).toProd.1 = (w (i + 1)).toProd.2) (x : Fin 2 → nonisolatedZigzagQuotient k G) : ((zigzagLoopAInfinityAlgebra hns c hc).m 2) x = x 0 * x 1

The binary operation of the loop deformation is the product of the zigzag algebra.

theorem TauCeti.zigzagLoopAInfinityAlgebra_m_eq_linearCombination {k : Type w} [CommRing k] {V : Type u} {G : SimpleGraph V} [Finite V] {hns : ∀ (i : V), ∃ (j : V), G.Adj i j} {n : ℕ} [NeZero n] (c : (Fin n → G.Dart) →₀ k) (hc : ∀ w ∈ c.support, ∀ (i : Fin n), (w i).toProd.1 = (w (i + 1)).toProd.2) (hn : 3 ≤ n) : (zigzagLoopAInfinityAlgebra hns c hc).m n = (Finsupp.linearCombination k (zigzagLoopOperation k G hns)) c

For n ≥ 3, the operation of arity n of the loop deformation is the linear combination of loop operations with coefficients c.

theorem TauCeti.zigzagLoopAInfinityAlgebra_m_mem {k : Type w} [CommRing k] {V : Type u} {G : SimpleGraph V} [Finite V] {hns : ∀ (i : V), ∃ (j : V), G.Adj i j} {n : ℕ} [NeZero n] (c : (Fin n → G.Dart) →₀ k) (hc : ∀ w ∈ c.support, ∀ (i : Fin n), (w i).toProd.1 = (w (i + 1)).toProd.2) (hn : 3 ≤ n) (y : Fin n → nonisolatedZigzagQuotient k G) : ((zigzagLoopAInfinityAlgebra hns c hc).m n) y ∈ zigzagIntegerGrade k G 2

The operation of arity n ≥ 3 of the loop deformation takes values of degree two, in the volume classes.

theorem TauCeti.zigzagLoopAInfinityAlgebra_m_eq_zero {k : Type w} [CommRing k] {V : Type u} {G : SimpleGraph V} [Finite V] {hns : ∀ (i : V), ∃ (j : V), G.Adj i j} {n : ℕ} [NeZero n] (c : (Fin n → G.Dart) →₀ k) (hc : ∀ w ∈ c.support, ∀ (i : Fin n), (w i).toProd.1 = (w (i + 1)).toProd.2) {m : ℕ} (h₂ : m ≠ 2) (hn : m ≠ n) : (zigzagLoopAInfinityAlgebra hns c hc).m m = 0

Every operation of the loop deformation other than m₂ and mₙ vanishes.

theorem TauCeti.zigzagLoopAInfinityAlgebra_m_ofArrow {k : Type w} [CommRing k] {V : Type u} {G : SimpleGraph V} [Finite V] {hns : ∀ (i : V), ∃ (j : V), G.Adj i j} {n : ℕ} [NeZero n] (c : (Fin n → G.Dart) →₀ k) (hc : ∀ w ∈ c.support, ∀ (i : Fin n), (w i).toProd.1 = (w (i + 1)).toProd.2) (hn : 3 ≤ n) (d : Fin n → G.Dart) : (((zigzagLoopAInfinityAlgebra hns c hc).m n) fun (i : Fin n) => (zigzagMk k G) (PathAlgebra.ofArrow (DoubledQuiver.arrow G ⋯))) = c d • zigzagVolume k G (d 0).toProd.2

The higher operation on arrows: for n ≥ 3, on the arrows of n darts d, the operation of arity n of the loop deformation is the coefficient c d times the volume class at the head of d 0.

theorem TauCeti.isMinimal_zigzagLoopAInfinityAlgebra {k : Type w} [CommRing k] {V : Type u} {G : SimpleGraph V} [Finite V] {hns : ∀ (i : V), ∃ (j : V), G.Adj i j} {n : ℕ} [NeZero n] (c : (Fin n → G.Dart) →₀ k) (hc : ∀ w ∈ c.support, ∀ (i : Fin n), (w i).toProd.1 = (w (i + 1)).toProd.2) : (zigzagLoopAInfinityAlgebra hns c hc).IsMinimal

The loop deformation is minimal: its unary operation vanishes.

theorem TauCeti.strictUnit_one_zigzagLoopAInfinityAlgebra {k : Type w} [CommRing k] {V : Type u} {G : SimpleGraph V} [Finite V] {hns : ∀ (i : V), ∃ (j : V), G.Adj i j} {n : ℕ} [NeZero n] (c : (Fin n → G.Dart) →₀ k) (hc : ∀ w ∈ c.support, ∀ (i : Fin n), (w i).toProd.1 = (w (i + 1)).toProd.2) : (zigzagLoopAInfinityAlgebra hns c hc).StrictUnit 1

The loop deformation is strictly unital, with the unit of the zigzag algebra as strict unit: the higher operation vanishes when an input is 1, which has no arrow coordinates.

theorem TauCeti.isHomogeneous_zigzagLoopAInfinityAlgebra_m_adams {k : Type w} [CommRing k] {V : Type u} {G : SimpleGraph V} [Finite V] {hns : ∀ (i : V), ∃ (j : V), G.Adj i j} {n : ℕ} [NeZero n] (c : (Fin n → G.Dart) →₀ k) (hc : ∀ w ∈ c.support, ∀ (i : Fin n), (w i).toProd.1 = (w (i + 1)).toProd.2) (r : ℕ) : MultilinearMap.IsHomogeneous ((zigzagLoopAInfinityAlgebra hns c hc).m r) (fun (x : Fin r) (q : ℤ) => zigzagIntegerGrade k G (-q)) (fun (q : ℤ) => zigzagIntegerGrade k G (-q)) (↑r - 2)

The Adams degree of the loop deformation: for the Adams grading, the negative of path length, in which an arrow has degree -1, the operation m r is homogeneous of degree r - 2. Together with its cohomological degree 2 - r, this is the bidegree (2 - r, r - 2) of m r when each arrow has bidegree (1, -1).

The strict zigzag algebra

noncomputable def TauCeti.zigzagStrictAInfinityAlgebra (k : Type w) [CommRing k] {V : Type u} {G : SimpleGraph V} [Finite V] (hns : ∀ (i : V), ∃ (j : V), G.Adj i j) : AInfinityAlgebra k (nonisolatedZigzagQuotient k G)

The strict zigzag algebra as an A∞ algebra: the loop deformation with zero coefficients, graded by path length, with m₂ the product and every other operation zero.

theorem TauCeti.zigzagStrictAInfinityAlgebra_def {k : Type w} [CommRing k] {V : Type u} {G : SimpleGraph V} [Finite V] {hns : ∀ (i : V), ∃ (j : V), G.Adj i j} : zigzagStrictAInfinityAlgebra k hns = zigzagLoopAInfinityAlgebra hns 0 ⋯

The strict zigzag algebra is the loop deformation with zero coefficients.

@[simp]

theorem TauCeti.zigzagStrictAInfinityAlgebra_grading_piece {k : Type w} [CommRing k] {V : Type u} {G : SimpleGraph V} [Finite V] {hns : ∀ (i : V), ∃ (j : V), G.Adj i j} : (zigzagStrictAInfinityAlgebra k hns).grading.piece ℤ = zigzagIntegerGrade k G

The strict zigzag algebra is graded by path length.

theorem TauCeti.zigzagLoopAInfinityAlgebra_grading_eq_zigzagStrictAInfinityAlgebra_grading {k : Type w} [CommRing k] {V : Type u} {G : SimpleGraph V} [Finite V] {hns : ∀ (i : V), ∃ (j : V), G.Adj i j} {n : ℕ} [NeZero n] (c : (Fin n → G.Dart) →₀ k) (hc : ∀ w ∈ c.support, ∀ (i : Fin n), (w i).toProd.1 = (w (i + 1)).toProd.2) : (zigzagLoopAInfinityAlgebra hns c hc).grading = (zigzagStrictAInfinityAlgebra k hns).grading

A loop deformation has the grading of the strict zigzag algebra.

@[simp]

theorem TauCeti.zigzagStrictAInfinityAlgebra_m_two_apply {k : Type w} [CommRing k] {V : Type u} {G : SimpleGraph V} [Finite V] {hns : ∀ (i : V), ∃ (j : V), G.Adj i j} (x : Fin 2 → nonisolatedZigzagQuotient k G) : ((zigzagStrictAInfinityAlgebra k hns).m 2) x = x 0 * x 1

The binary operation of the strict zigzag algebra is the product.

theorem TauCeti.zigzagLoopAInfinityAlgebra_m_two_eq_zigzagStrictAInfinityAlgebra_m_two {k : Type w} [CommRing k] {V : Type u} {G : SimpleGraph V} [Finite V] {hns : ∀ (i : V), ∃ (j : V), G.Adj i j} {n : ℕ} [NeZero n] (c : (Fin n → G.Dart) →₀ k) (hc : ∀ w ∈ c.support, ∀ (i : Fin n), (w i).toProd.1 = (w (i + 1)).toProd.2) : (zigzagLoopAInfinityAlgebra hns c hc).m 2 = (zigzagStrictAInfinityAlgebra k hns).m 2

A loop deformation has the binary operation of the strict zigzag algebra.

theorem TauCeti.zigzagStrictAInfinityAlgebra_m_eq_zero {k : Type w} [CommRing k] {V : Type u} {G : SimpleGraph V} [Finite V] {hns : ∀ (i : V), ∃ (j : V), G.Adj i j} {m : ℕ} (hm : m ≠ 2) : (zigzagStrictAInfinityAlgebra k hns).m m = 0

The strict zigzag algebra has no operation besides the product.

theorem TauCeti.isMinimal_zigzagStrictAInfinityAlgebra {k : Type w} [CommRing k] {V : Type u} {G : SimpleGraph V} [Finite V] {hns : ∀ (i : V), ∃ (j : V), G.Adj i j} : (zigzagStrictAInfinityAlgebra k hns).IsMinimal

The strict zigzag algebra is minimal.