Gauge-trivial quartic loop deformations of a zigzag algebra #
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). For two darts d and e with d.fst = e.snd, the
backtrack operation
ψ_{d,e}(x₀, x₁, x₂) = a_d(x₀) a_e(x₁) a_{e.symm}(x₂) · d,
with a_d the coordinate of the arrow of d in the vertex-arrow-volume basis, is a ternary
operation of degree -2. On arrows it is nonzero only on the word d, e, e.symm, which it sends to
d: the backtrack along e is removed. Its Hochschild coboundary
x₀ ψ(x₁, x₂, x₃) - ψ(x₀ x₁, x₂, x₃) + ψ(x₀, x₁ x₂, x₃) - ψ(x₀, x₁, x₂ x₃) + ψ(x₀, x₁, x₂) x₃
is the sum of the loop operations TauCeti.zigzagLoopOperation of the closed walk
d.symm, d, e, e.symm and of its rotation d, e, e.symm, d.symm
(TauCeti.zigzagBacktrackOperation_coboundary). The proof is the telescoping of arrow coordinates
used for the cocycle property of loop operations.
Let P be a finite set of such pairs, with weights a, and suppose that no first dart of a pair is
reverse to the first dart of another, so that the products of two values of the weighted
backtrack cochain ψ = ∑ a_p ψ_p vanish. Then the loop deformation
TauCeti.zigzagLoopAInfinityAlgebra whose coefficients are the weighted backtrack walks and their
rotations is gauge trivial. The maps f₁ = id, f₃ = ψ and fₙ = 0 otherwise are the
components of an A∞ morphism to the strict zigzag algebra (TauCeti.zigzagBacktrackAInfinityHom).
Its morphism equations are f₁ m₄ = δψ in arity four, m₂(f₃, f₃) = 0 in arity six, and trivial
otherwise, since ψ vanishes on the volume classes in which m₄ takes its values. The linear part
is the identity, so the morphism is an A∞ isomorphism and the deformation is formal.
The signs are those of Tau Ceti's Keller/Getzler--Jones conventions, in which the arity-four
morphism equation carries no relative sign between the two outer terms of δψ. Liu--Wang's quartic
operation for affine D₄, with equal values on the four rotations of its walk, is an instance;
see TauCeti.RepresentationTheory.Quiver.Zigzag.AInfinity.Affine.D.Four.
Main definitions #
TauCeti.zigzagBacktrackOperation: the backtrack operation of two darts.TauCeti.zigzagBacktrackWalk: the closed walkd.symm, d, e, e.symm.TauCeti.zigzagBacktrackCochain: a weighted sum of backtrack operations.TauCeti.zigzagBacktrackCoefficients: the coefficients of the corresponding loop deformation.TauCeti.zigzagBacktrackTaylor: the suspension of the backtrack cochain.TauCeti.zigzagBacktrackAInfinityHom: theA∞isomorphism to the strict zigzag algebra.
Main results #
TauCeti.zigzagBacktrackOperation_coboundaryandTauCeti.zigzagBacktrackCochain_coboundary: the quartic operation is the Hochschild coboundary of the backtrack cochain.TauCeti.taylor_zigzagBacktrackAInfinityHom,TauCeti.component_three_zigzagBacktrackAInfinityHomandTauCeti.component_zigzagBacktrackAInfinityHom_of_ne: the Taylor map of the backtrack isomorphism, its arity-three component, the backtrack cochain, and the vanishing of its components outside arities one and three.TauCeti.isIso_zigzagBacktrackAInfinityHomandTauCeti.isFormal_zigzagLoopAInfinityAlgebra_zigzagBacktrackCoefficients: the loop deformation with backtrack coefficients isA∞isomorphic to the strict zigzag algebra, hence formal.
References #
- B. Keller, Introduction to A-infinity algebras and modules, Section 3.4, for
A∞morphisms. - Y. Liu and Z. Wang, A-infinity deformations of zigzag algebras via Ginzburg dg algebras, Section 4 and Example 4.7.
Backtrack operations #
The backtrack operation of two darts d and e: the product of the d-, e- and
e.symm-arrow coordinates of its three inputs, times the arrow of d.
Instances For
A backtrack operation is the product of three arrow coordinates times an arrow.
The closed walk d.symm, d, e, e.symm in product order: read from the last entry it goes
back and forth along e and then along d, so it is closed when d.fst = e.snd.
Instances For
Every rotation of the backtrack walk of d and e is closed, read in product order, when d
starts where e ends.
The core coboundary identity. When d.fst = e.snd, the Hochschild coboundary of the
backtrack operation of d and e is the sum of the loop operations of the backtrack walk and of
its rotation by one step.
Weighted families of backtrack operations #
The backtrack cochain of a finite set P of pairs of darts with weights a: the sum of the
backtrack operations of the pairs in P, weighted by a.
Instances For
The backtrack cochain is a weighted sum of products of three arrow coordinates times arrows.
The coefficients of the loop operations of the backtrack walks of the pairs in P and of their
rotations by one step, weighted by a.
Instances For
The backtrack coefficients are the sum, over the pairs in P, of the single coefficients of the
backtrack walk and of its rotation by one step.
The backtrack coefficients are supported on closed walks, read in product order.
The quartic operation of the loop deformation with backtrack coefficients is the weighted sum of the loop operations of the backtrack walks and of their rotations.
The coboundary of the backtrack cochain is the quartic operation of the loop deformation with backtrack coefficients.
The backtrack cochain vanishes as soon as one input is homogeneous of degree other than one: it only reads arrow coordinates.
The backtrack cochain has degree -2 for the path-length grading.
Products of values of the backtrack cochain vanish when no pair in P has a first dart reverse
to the first dart of another.
The A∞ isomorphism #
The backtrack Taylor map: the suspension of the backtrack cochain, concentrated on words of length three. Together with the projection onto single letters it is the Taylor map of the backtrack isomorphism.
Instances For
The backtrack Taylor map vanishes on words whose length is not three.
On a word of length three the backtrack Taylor map is the negative of the backtrack cochain: the suspension sign of an arity-three operation of even degree is that of its middle letter, which the backtrack cochain only reads through an arrow coordinate.
The backtrack Taylor map has degree zero for the suspended path-length grading.
The backtrack isomorphism: the A∞ morphism from the loop deformation with backtrack
coefficients to the strict zigzag algebra whose Taylor map is the projection onto single letters
plus the backtrack Taylor map. In unsuspended terms its components are f₁ = id, f₃ the
backtrack cochain, and fₙ = 0 otherwise.
Instances For
The Taylor map of the backtrack isomorphism.
The linear part of the backtrack isomorphism is the identity.
The arity-three component of the backtrack isomorphism is the backtrack cochain.
The components of the backtrack isomorphism vanish outside arities one and three.
The backtrack isomorphism is an A∞ isomorphism.
The loop deformation with backtrack coefficients is formal: it is A∞ isomorphic to the
strict zigzag algebra, which has no operation besides the product.