Non-formal quartic deformations of zigzag algebras #
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 quartic loop deformation
TauCeti.zigzagLoopAInfinityAlgebra is a minimal, strictly unital A∞ structure on Z with the
same grading and product, m₃ = 0, and m₄ a linear combination of loop operations of closed
walks. This file shows that such deformations can be nontrivial modulo A∞ isomorphism, and
hence that Z is in general not intrinsically formal.
This combines two results. First, when 2 is invertible, a minimal A∞
structure with m₃ = 0 which is A∞ isomorphic to the graded algebra it deforms has a quartic
operation which is a Hochschild coboundary, even when the isomorphism has a nonzero quadratic
component
(IsMinimal.not_isFormal_of_not_exists_hochschildDifferential_eq_obstruction_four in
TauCeti.AInfinityAlgebra).
Second, the loop operation of a single closed walk of the shape n₁ → v → n₀ → v → n₁ is not a
Hochschild coboundary in two situations: when the vertex v has four distinct neighbours, two of
which are n₀ and n₁
(TauCeti.not_exists_hochschildDifferential_eq_obstruction_zigzagLoopAInfinityAlgebra_single),
and when v has three distinct neighbours, two of which are n₀ and n₁, and is joined through
the third by a suitable walk to another vertex with three distinct neighbours
(not_exists_hochschildDifferential_eq_obstruction_zigzagLoopAInfinityAlgebra_single_of_walk).
Hence the loop deformation with coefficient one on that walk is not formal, equivalently not
A∞ isomorphic to the strict zigzag algebra, and the strict zigzag algebra is not intrinsically
formal. For the affine D₄ graph the walk is Liu--Wang's β₄ α₁ β₁ α₄; see
TauCeti.RepresentationTheory.Quiver.Zigzag.AInfinity.Affine.D.Four.
In terms of degrees, the strict zigzag algebra is not intrinsically formal when G has a vertex of
degree at least four, or two distinct vertices of degree at least three in one connected
component; for the second criterion the walk is taken along a path between the two vertices. The
results assume that 2 is invertible, as the invariance of the quartic obstruction does. Among the
finite trees which are not Dynkin diagrams they cover all except those with exactly one vertex of
degree three and none of larger degree, such as the affine E₆, E₇ and E₈ graphs; in
particular they cover the affine Dₙ graphs, see
TauCeti.RepresentationTheory.Quiver.Zigzag.AInfinity.Affine.D.Basic.
Main results #
TauCeti.not_isFormal_zigzagLoopAInfinityAlgebra: when2is invertible, a loop deformation whose quartic operation is not a Hochschild coboundary of the strict zigzag algebra is not formal.TauCeti.not_isIntrinsicallyFormal_zigzagStrictAInfinityAlgebra_of_not_isFormal: if a loop deformation is not formal, then the strict zigzag algebra is not intrinsically formal.TauCeti.not_isFormal_zigzagLoopAInfinityAlgebra_singleandTauCeti.not_isFormal_zigzagLoopAInfinityAlgebra_single_of_walk: the loop deformation of a single closed walkn₁ → v → n₀ → v → n₁is not formal, at a vertex with four distinct neighbours, or at a vertex with three distinct neighbours joined by a walk to another such vertex.TauCeti.not_isIntrinsicallyFormal_zigzagStrictAInfinityAlgebraandTauCeti.not_isIntrinsicallyFormal_zigzagStrictAInfinityAlgebra_of_walk: the corresponding statements that the strict zigzag algebra is not intrinsically formal.TauCeti.not_isIntrinsicallyFormal_zigzagStrictAInfinityAlgebra_of_four_le_degreeandTauCeti.not_isIntrinsicallyFormal_zigzagStrictAInfinityAlgebra_of_three_le_degree: the strict zigzag algebra is not intrinsically formal whenGhas a vertex of degree at least four, or two distinct vertices of degree at least three joined by a walk.
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 their intrinsic formality. - T. Kadeishvili, The structure of the
A(∞)-algebra, and the Hochschild and Harrison cohomologies, Trudy Tbiliss. Mat. Inst. 91 (1988), 19--27, for the obstruction method.
A loop deformation whose quartic operation is not a coboundary is not formal, when 2 is
invertible. If the quartic operation of the loop deformation with coefficients c is not the
Hochschild differential of a suspended cochain of degree zero of the strict zigzag algebra, then
the loop deformation is not formal. Equivalently it is not A∞ isomorphic to the strict zigzag
algebra (TauCeti.AInfinityAlgebra.IsMinimal.isFormal_iff_exists_isIso).
A non-formal loop deformation shows that the strict zigzag algebra is not intrinsically
formal. A loop deformation is a minimal A∞ structure on the zigzag algebra with its grading and
product, so if it is not formal, then the strict zigzag algebra is not intrinsically formal.
A non-formal quartic deformation of a zigzag algebra. Assume that 2 is invertible and
that the ring is nontrivial. Let δ 0, …, δ 3 be four distinct darts with a common head v, for
instance the darts into v from four distinct neighbours. The loop deformation whose quartic
operation is the loop operation of the single closed walk (δ 3).symm, δ 0, (δ 0).symm, δ 3 (in
product order) is a minimal, strictly unital A∞ structure on the zigzag algebra, with its grading
and product, which is not formal: it is not A∞ isomorphic to the strict zigzag algebra.
The zigzag algebra of a graph with a vertex of valency at least four is not intrinsically
formal, over every nontrivial commutative ring in which 2 is invertible: when four distinct
darts of G have a common head, some A∞ algebra whose cohomology algebra is the path-graded
zigzag algebra is not formal.
The zigzag algebra of a graph with a vertex of degree at least four is not intrinsically
formal, over every nontrivial commutative ring in which 2 is invertible. This is
TauCeti.not_isIntrinsicallyFormal_zigzagStrictAInfinityAlgebra for the darts into v from four
distinct neighbours.
Two branch vertices joined by a walk #
A non-formal quartic deformation at two branch vertices. Assume that 2 is invertible and
that the ring is nontrivial. Let d and e be distinct darts with a common head v, and let the
non-backtracking walk of darts f 0, …, f n lead from v, along neither d.symm nor e.symm,
to a vertex w without returning to v. Let d' and e' be distinct darts with head w, both
distinct from the last dart f n of the walk. The loop deformation whose quartic operation is the
loop operation of the single closed walk e.symm, d, d.symm, e (in product order) is a minimal,
strictly unital A∞ structure on the zigzag algebra, with its grading and product, which is not
formal: it is not A∞ isomorphic to the strict zigzag algebra.
The zigzag algebra of a graph with two branch vertices joined by a suitable walk is not
intrinsically formal, over every nontrivial commutative ring in which 2 is invertible. The
hypotheses are those of TauCeti.not_isFormal_zigzagLoopAInfinityAlgebra_single_of_walk: two
vertices with three distinct neighbours each, joined by a non-backtracking walk which leaves the
first through its third neighbour, enters the second through its third neighbour, and does not
return to the first.
The zigzag algebra of a graph with two branch vertices in one component is not
intrinsically formal, over every nontrivial commutative ring in which 2 is invertible: if two
distinct vertices v and w of G, each of degree at least three, are joined by a walk, then
some A∞ algebra whose cohomology algebra is the path-graded zigzag algebra is not formal. The
detecting deformation is that of TauCeti.not_isFormal_zigzagLoopAInfinityAlgebra_single_of_walk
along a path from v to w. Together with
TauCeti.not_isIntrinsicallyFormal_zigzagStrictAInfinityAlgebra_of_four_le_degree this covers
every finite tree other than a path or a tree with exactly one vertex of degree three.