Liu--Wang's quartic A∞ deformation of the affine D₄ zigzag algebra #
Liu and Wang (Example 4.7) exhibit a minimal A∞ structure on the zigzag algebra of the affine
D₄ graph whose only higher operation is a quartic operation m₄ built from one closed walk. In
the numbering of TauCeti.AffineDynkinType.graph (D 4), node 1 is the centre, of valency four,
and 0, 2, 3, 4 are leaves. Write α_l for the arrow from a leaf l to the centre and β_l for
the arrow from the centre to l. Liu--Wang's two leaves 1 and 4 are taken to be the nodes 0
and 4. For one common scalar c, the operation m₄ vanishes on every arrow quadruple other than
the four cyclic rotations of the closed walk β₄ α₁ β₁ α₄, and on those it takes the values
m₄(β₄, α₁, β₁, α₄) = c β₄ α₄, m₄(α₄, β₄, α₁, β₁) = c α₁ β₁,
m₄(β₁, α₄, β₄, α₁) = c β₁ α₁, m₄(α₁, β₁, α₄, β₄) = c α₄ β₄.
Products are in Tau Ceti's later-factor-first order, so the last input of each word is traversed
first, and each value is the volume class at the base of the walk. Each rotation of the walk gives
a loop operation TauCeti.zigzagLoopOperation, and m₄ is c times their sum. The general
construction TauCeti.zigzagLoopAInfinityAlgebra therefore proves the Stasheff identities, with
arrows in cohomological degree 1. It also shows that the resulting algebra is minimal and
strictly unital, and that m₄ has bidegree (-2, 2) for the cohomological and Adams gradings.
With equal values on the four rotations, as displayed, this operation is a Hochschild coboundary
of the strict zigzag algebra in Tau Ceti's sign conventions. The four rotations of the walk are the
backtrack walks of the pairs (β₁, α₄) and (β₄, α₁) and their rotations by one step, so m₄ is
the coboundary of c times the backtrack cochain of these pairs, which sends (β₁, α₄, β₄) to
β₁ and (β₄, α₁, β₁) to β₄ (TauCeti.zigzagBacktrackOperation_coboundary). Hence Liu--Wang's
algebra is A∞ isomorphic to the strict zigzag algebra, through an isomorphism whose linear part
is the identity, and it is formal (TauCeti.isFormal_liuWangAffineD4AInfinityAlgebra). In these
conventions the displayed values therefore define the trivial A∞ deformation.
A single rotation is not a coboundary. The loop operation of the walk β₄ α₁ β₁ α₄ alone, with
coefficient one, is a Hochschild cocycle
(TauCeti.hochschildDifferential_obstruction_zigzagLoopAInfinityAlgebra) which is not a
coboundary, since the centre has four distinct neighbours
(TauCeti.not_exists_hochschildDifferential_eq_obstruction_liuWangAffineD4Walk); see
TauCeti.RepresentationTheory.Quiver.Zigzag.AInfinity.HochschildClass. In particular HH^{4, -2}
of the affine D₄ zigzag algebra is nonzero (TauCeti.not_hochschildVanishesAt_four_affineD4).
When 2 is invertible, the loop deformation of the single walk is therefore not formal, that is
not A∞ isomorphic to the strict zigzag algebra
(TauCeti.not_isFormal_zigzagLoopAInfinityAlgebra_liuWangAffineD4Walk), and the affine D₄
zigzag algebra is not intrinsically formal (TauCeti.not_isIntrinsicallyFormal_affineD4); see
TauCeti.RepresentationTheory.Quiver.Zigzag.AInfinity.NonFormal.
Main definitions #
TauCeti.liuWangAffineD4Walk: the closed walkβ₄ α₁ β₁ α₄, as four darts in product order.TauCeti.liuWangAffineD4AInfinityAlgebra: Liu--Wang'sA∞algebra with scalarc.TauCeti.liuWangAffineD4BacktrackPairs: the pairs(β₁, α₄)and(β₄, α₁), whose backtrack cochain boundsm₄.
Main results #
TauCeti.liuWangAffineD4AInfinityAlgebra_m_four_values: the four displayed values ofm₄.TauCeti.liuWangAffineD4AInfinityAlgebra_m_four_eq_zero:m₄vanishes on every other arrow quadruple.TauCeti.liuWangAffineD4AInfinityAlgebra_m_eq_zero: there are no other higher operations.TauCeti.isMinimal_liuWangAffineD4AInfinityAlgebra,TauCeti.strictUnit_one_liuWangAffineD4AInfinityAlgebraandTauCeti.isHomogeneous_liuWangAffineD4AInfinityAlgebra_m_adams: the algebra is minimal and strictly unital, andm₄has bidegree(-2, 2).TauCeti.exists_isIso_liuWangAffineD4AInfinityAlgebraandTauCeti.isFormal_liuWangAffineD4AInfinityAlgebra: the algebra isA∞isomorphic to the strict zigzag algebra, and formal.TauCeti.not_exists_hochschildDifferential_eq_obstruction_liuWangAffineD4WalkandTauCeti.not_hochschildVanishesAt_four_affineD4: the loop operation of the walk alone is not a Hochschild coboundary, soHH^{4, -2}of the affineD₄zigzag algebra is nonzero.TauCeti.not_isFormal_zigzagLoopAInfinityAlgebra_liuWangAffineD4WalkandTauCeti.not_isIntrinsicallyFormal_affineD4: when2is invertible, the loop deformation of the walk alone is not formal, so the affineD₄zigzag algebra is not intrinsically formal.
References #
- Y. Liu and Z. Wang, A-infinity deformations of zigzag algebras via Ginzburg dg algebras, Example 4.7.
Liu--Wang's closed walk β₄ α₁ β₁ α₄ in the affine D₄ graph, listed in product order:
the darts 1 → 4, 0 → 1, 1 → 0 and 4 → 1. Read from the last entry, it is the walk
4 → 1 → 0 → 1 → 4.
Instances For
The darts of Liu--Wang's walk are pairwise distinct.
Consecutive darts of every cyclic rotation of Liu--Wang's walk are composable, cyclically.
The class in the affine D₄ zigzag algebra of the arrow of the i-th dart of Liu--Wang's walk:
β₄, α₁, β₁ and α₄ for i = 0, 1, 2, 3.
Instances For
The coefficients of Liu--Wang's quartic operation: the scalar c on each of the four cyclic
rotations of the walk, and zero elsewhere.
Instances For
The coefficient of four darts is c exactly when they form a cyclic rotation of Liu--Wang's
walk.
Liu--Wang's coefficients are supported on closed walks, read in product order.
Liu--Wang's A∞ algebra on the affine D₄ zigzag algebra, with scalar c: the
path-length graded zigzag algebra with m₂ the product and m₄ equal to c times the sum of the
loop operations of the four cyclic rotations of the walk β₄ α₁ β₁ α₄.
Instances For
The binary operation of Liu--Wang's algebra is the product of the zigzag algebra.
On a cyclic rotation of Liu--Wang's walk, m₄ is c times the volume class at the base of the
rotated walk.
The four values of Liu--Wang's m₄, in the order of Example 4.7:
m₄(β₄, α₁, β₁, α₄) = c β₄ α₄, m₄(α₄, β₄, α₁, β₁) = c α₁ β₁, m₄(β₁, α₄, β₄, α₁) = c β₁ α₁ and
m₄(α₁, β₁, α₄, β₄) = c α₄ β₄, where β₄, α₁, β₁, α₄ are the arrows
TauCeti.liuWangAffineD4Arrow 0, 1, 2, 3.
m₄ vanishes on every other arrow quadruple: on the arrows of four darts that do not form
a cyclic rotation of Liu--Wang's walk, the quartic operation is zero.
Liu--Wang's algebra is minimal.
Liu--Wang's algebra is strictly unital, with the unit of the zigzag algebra as strict unit.
The bidegree of Liu--Wang's operations: for the Adams grading, the negative of path length,
m n is homogeneous of degree n - 2. With its cohomological degree 2 - n, m₄ has bidegree
(-2, 2) when each arrow has bidegree (1, -1).
Formality #
The two pairs of darts whose backtrack operations bound Liu--Wang's quartic operation:
(β₁, α₄) and (β₄, α₁).
Instances For
In each of Liu--Wang's backtrack pairs, the first dart starts where the second ends.
No first dart of one of Liu--Wang's backtrack pairs is reverse to the first dart of another.
Liu--Wang's coefficients are backtrack coefficients: the four cyclic rotations of the walk
β₄ α₁ β₁ α₄ are the backtrack walks of (β₁, α₄) and (β₄, α₁) and their rotations by one
step.
Liu--Wang's A∞ algebra is A∞ isomorphic to the strict zigzag algebra, through an
isomorphism with linear part the identity: the quartic operation is the Hochschild coboundary of
the backtrack cochain of (β₁, α₄) and (β₄, α₁), scaled by c
(TauCeti.zigzagBacktrackAInfinityHom).
Liu--Wang's A∞ algebra is formal, for every scalar c: the displayed quartic operation
m₄ is a Hochschild coboundary of the strict zigzag algebra, so it defines the trivial
A∞ deformation.
A single rotation of the walk #
A single coefficient on Liu--Wang's walk is supported on closed walks, read in product order.
The loop operation of Liu--Wang's walk alone is not a Hochschild coboundary. With
coefficient one on the single walk β₄ α₁ β₁ α₄, and zero on its other rotations, the quartic
operation of the loop deformation is a Hochschild cocycle of the strict zigzag algebra
(TauCeti.hochschildDifferential_obstruction_zigzagLoopAInfinityAlgebra), and it is not the
Hochschild differential of any suspended cochain of degree zero. By contrast the sum of the four
rotations with equal coefficients, Liu--Wang's operation, is a coboundary
(TauCeti.isFormal_liuWangAffineD4AInfinityAlgebra).
The quartic Hochschild cohomology HH^{4, -2} of the affine D₄ zigzag algebra is
nonzero, over every nontrivial commutative ring: this is the case G = D₄ of
TauCeti.not_hochschildVanishesAt_four_zigzagStrictAInfinityAlgebra_of_isContained.
The loop operation of Liu--Wang's walk alone gives a non-formal deformation, over every
nontrivial commutative ring in which 2 is invertible: the loop deformation with coefficient one
on the single walk β₄ α₁ β₁ α₄, and zero on its other rotations, is a minimal, strictly unital
A∞ structure on the affine D₄ zigzag algebra, with its grading and product, which is not
formal, that is not A∞ isomorphic to the strict zigzag algebra.
The affine D₄ zigzag algebra is not intrinsically formal, over every nontrivial
commutative ring in which 2 is invertible: the centre has the four distinct neighbours 0, 2,
3 and 4.