Nonzero quartic Hochschild classes 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). Its Hochschild cohomology HH^{4, -2}(Z, Z) is the
obstruction group for the quartic operations of minimal A∞ structures on Z
(TauCeti.AInfinityAlgebra.HochschildVanishesAt). This file shows that it is nonzero, over every
nontrivial commutative ring, as soon as some vertex v of G has four distinct neighbours, for
example the centre of the affine D₄ graph, or two vertices of G with three distinct neighbours
each are joined by a walk of a suitable shape, for example the two branch vertices of the affine
Dₙ graphs for n ≥ 5. The class is that of the loop operation TauCeti.zigzagLoopOperation of
a single closed walk which goes from a neighbour of v to v, makes a detour to another neighbour
and back, and returns.
The detecting cycle #
Since Z is a symmetric algebra, a Hochschild cochain can be paired with a Hochschild chain by
means of the trace TauCeti.zigzagTrace. Fix a symmetric matrix a of weights on pairs of darts
which vanishes on the pairs of darts with distinct heads. Write B d e for the backtrack walk
TauCeti.zigzagBacktrackWalk G e.symm d, that is the darts e, e.symm, d, d.symm in product
order, and B' d e for its rotation e.symm, d, d.symm, e. The functional
Λ(F) = ∑ d e, a d e (tr F(B' d e) - tr F(B d e)),
evaluated on the words of the corresponding arrows (TauCeti.zigzagBacktrackPairing), vanishes on
every Hochschild coboundary of degree zero as soon as the weights satisfy the balance condition
∑ d, a d e = a e.symm e.symm for every dart e
(TauCeti.zigzagBacktrackPairing_hochschildDifferential_eq_zero).
On a word of four arrows the Hochschild differential of a suspended cochain φ is
δφ(x₀, x₁, x₂, x₃) = φ(x₀x₁, x₂, x₃) + φ(x₀, x₁x₂, x₃) + φ(x₀, x₁, x₂x₃)
- φ(x₀, x₁, x₂) x₃ - x₀ φ(x₁, x₂, x₃)
(TauCeti.zigzagStrictAInfinityAlgebra_hochschildDifferential_apply_of_tprod), and on the words
above the only nonzero products of consecutive arrows are backtracks, which are volume classes. For
distinct darts d and e with head v, the outer terms of B' d e cancel against those of
B d e and B e d by the trace property tr(xy) = tr(yx) and the symmetry of a, leaving terms
containing the volume class x_v. For d = e the two words are the double backtracks
d.symm, d, d.symm, d and d, d.symm, d, d.symm, which leave terms containing the volume classes
at both ends of d; this is why the diagonal weight of e.symm enters the balance condition at
e. The terms containing volume classes then cancel by the balance condition.
The balance condition holds in two situations.
Four distinct darts
δ iwith a common head, with the weightsa = [ 0 -1 0 1] [-1 0 1 0] [ 0 1 0 -1] [ 1 0 -1 0]on the pairs
(δ i, δ j), which have zero diagonal and zero row sums.Distinct darts
dandewith a common headv, joined by a non-backtracking walk of dartsf 0, …, f nto distinct dartsd'ande'with a common headw. The weights are1on the pair{d, e}and-1on the pairs{d, (f 0).symm}and{e, (f 0).symm}, then2 (-1)^ton the diagonal entry of(f t).symmand on the pair{f t, (f (t + 1)).symm}, and finally(-1)^(n + 1)times the weights1, -1, -1on the pairs{d', e'},{d', f n},{e', f n}. The balance condition along the walk is a telescoping identity.
On the other hand Λ pairs the obstruction of the loop deformation
TauCeti.zigzagLoopAInfinityAlgebra with coefficients c, its operation m₄, to
∑ d e, a d e (c (B' d e) - c (B d e)) (TauCeti.zigzagBacktrackPairing_obstruction). For the
single walk B' d e with coefficient one, for distinct darts d and e, this is a d e, which
is 1 for the pair (δ 0, δ 3) in the first situation and the pair (d, e) in the second. So its
loop operation, a Hochschild cocycle
(TauCeti.hochschildDifferential_obstruction_zigzagLoopAInfinityAlgebra), is not a coboundary
(TauCeti.not_exists_hochschildDifferential_eq_obstruction_zigzagLoopAInfinityAlgebra_single and
not_exists_hochschildDifferential_eq_obstruction_zigzagLoopAInfinityAlgebra_single_of_walk),
and HH^{4, -2}(Z, Z) ≠ 0. In particular this holds for the affine Dₙ graphs with n ≥ 5
(TauCeti.not_hochschildVanishesAt_four_affineD), as it does for the affine D₄ graph
(TauCeti.not_hochschildVanishesAt_four_affineD4); see
TauCeti.RepresentationTheory.Quiver.Zigzag.AInfinity.Affine.D.Basic and
TauCeti.RepresentationTheory.Quiver.Zigzag.AInfinity.Affine.D.Four.
This is consistent with TauCeti.zigzagBacktrackOperation_coboundary, by which the sum of the loop
operations of a backtrack walk B d e and of its rotation B' d e is a coboundary: Λ takes
opposite values on them. In particular the four rotations of one such walk with equal coefficients,
as in Liu--Wang's affine D₄ example, have a coboundary as their sum, while a single rotation does
not.
Kadeishvili's criterion
TauCeti.AInfinityAlgebra.isIntrinsicallyFormal_of_forall_hochschildVanishesAt therefore does not
apply to such zigzag algebras. That the corresponding loop deformation is not A∞ isomorphic to
the strict zigzag algebra, when 2 is invertible, is proved in
TauCeti.RepresentationTheory.Quiver.Zigzag.AInfinity.NonFormal.
Main definitions #
TauCeti.zigzagBacktrackPairing: the functionalΛon suspended Hochschild cochains.
Main results #
TauCeti.zigzagStrictAInfinityAlgebra_hochschildDifferential_apply_of_tprod_of_mem: the Hochschild differential of a degree-zero suspended cochain on a word of letters of path length one, of any length at least two.TauCeti.zigzagStrictAInfinityAlgebra_hochschildDifferential_apply_of_tprod: the same on a word of four letters, written out.TauCeti.zigzagBacktrackPairing_hochschildDifferential_eq_zero:Λvanishes on coboundaries when its weights satisfy the balance condition.TauCeti.zigzagTrace_obstruction_zigzagLoopAInfinityAlgebra_of_tprod: the trace of the higher operation of a loop deformation on a word of arrows, in every arity.TauCeti.zigzagBacktrackPairing_obstruction: the value ofΛon the quartic operation of a loop deformation.TauCeti.hochschildDifferential_obstruction_zigzagLoopAInfinityAlgebra: the higher operation of a loop deformation is a Hochschild cocycle, in every arity.TauCeti.not_hochschildVanishesAt_zigzagStrictAInfinityAlgebra_of_not_exists_eq_obstruction: a higher loop operation which is not a coboundary gives a nonzero class inHH^{n + 2, -n}(Z, Z).TauCeti.not_exists_hochschildDifferential_eq_obstruction_zigzagLoopAInfinityAlgebraandTauCeti.not_exists_hochschildDifferential_eq_obstruction_zigzagLoopAInfinityAlgebra_single: quartic loop operations which are not Hochschild coboundaries.not_exists_hochschildDifferential_eq_obstruction_zigzagLoopAInfinityAlgebra_single_of_walk: the same at a vertex joined by a walk to another vertex, both with three distinct neighbours.TauCeti.not_hochschildVanishesAt_four_zigzagStrictAInfinityAlgebraandTauCeti.not_hochschildVanishesAt_four_zigzagStrictAInfinityAlgebra_of_walk:HH^{4, -2}(Z, Z) ≠ 0when a vertex ofGhas four distinct neighbours, or two vertices with three distinct neighbours each are joined by a suitable 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 Hochschild cohomology. - J.-L. Loday, Cyclic Homology, Chapter 1, for Hochschild chains and cochains and their duality.
The Hochschild differential on words of four arrows #
The Hochschild differential on words of arrows of any length. For a suspended cochain φ
of degree zero and letters x₀, …, x_{n+1} of path length one, such as arrows,
δφ(x₀, …, x_{n+1}) = ∑_{p ≤ n} φ(x₀, …, x_p x_{p+1}, …, x_{n+1})
- φ(x₀, …, x_n) x_{n+1} - x₀ φ(x₁, …, x_{n+1}).
The quartic case, with the words written out, is
zigzagStrictAInfinityAlgebra_hochschildDifferential_apply_of_tprod.
The Hochschild differential on words of four letters of path length one. For a suspended
cochain φ of degree zero and letters x₀, x₁, x₂, x₃ of path length one, such as arrows,
δφ(x₀, x₁, x₂, x₃) = φ(x₀x₁, x₂, x₃) + φ(x₀, x₁x₂, x₃) + φ(x₀, x₁, x₂x₃)
- φ(x₀, x₁, x₂) x₃ - x₀ φ(x₁, x₂, x₃).
The pairing with a Hochschild cycle #
The backtrack pairing with weights a d e on pairs of darts: the linear functional on
suspended Hochschild cochains F of the zigzag algebra
Λ(F) = ∑ d e, a d e (tr F(B' d e) - tr F(B d e)),
where B d e is the word of the arrows of the backtrack walk
TauCeti.zigzagBacktrackWalk G e.symm d, namely e, e.symm, d, d.symm, and B' d e that of its
rotation by one step, namely e.symm, d, d.symm, e. It is the pairing, through the trace, with the
Hochschild chain ∑ d e, a d e (1 ⊗ B' d e - 1 ⊗ B d e).
Instances For
Unfolding the backtrack pairing.
The backtrack pairing vanishes on Hochschild coboundaries. Let a be a symmetric matrix of
weights on pairs of darts which vanishes on pairs of darts with distinct heads, and whose column
sums satisfy the balance condition ∑ d, a d e = a e.symm e.symm at every dart e. Then the
backtrack pairing with weights a vanishes on the Hochschild differential of every suspended
cochain of degree zero.
The higher operation of a loop deformation under the trace. For n ≥ 1 and coefficients
c on walks of length n + 2, the trace of the obstruction of the loop deformation, its operation
of arity n + 2, on the arrows of n + 2 darts w is the coefficient of w times the sign
∏ i, (-1) ^ (n + 1 - i) of the suspension.
The backtrack pairing of a quartic loop operation. On the obstruction of the loop
deformation with coefficients c, that is on its quartic operation, the backtrack pairing is
∑ d e, a d e (c (B' d e) - c (B d e)), for B d e the backtrack walk of e.symm and d and
B' d e its rotation by one step.
Higher loop operations are Hochschild cocycles. For n ≥ 1, the obstruction of the loop
deformation with coefficients c on walks of length n + 2, that is its operation of arity
n + 2, is a Hochschild cocycle of the strict zigzag algebra, since the two agree up to arity
n + 1.
A higher loop operation which is not a coboundary gives a nonzero Hochschild class. If the
operation of arity n + 2 of a loop deformation, a Hochschild cocycle of the strict zigzag algebra
by TauCeti.hochschildDifferential_obstruction_zigzagLoopAInfinityAlgebra, is not the Hochschild
differential of any suspended cochain of degree zero, then HH^{n + 2, -n}(Z, Z) ≠ 0.
Quartic loop operations which are not Hochschild coboundaries. Let a be a symmetric
matrix of weights on pairs of darts which vanishes on pairs of darts with distinct heads and
satisfies the balance condition ∑ d, a d e = a e.symm e.symm. If the backtrack pairing of the
coefficients c of a loop deformation is nonzero, its quartic operation, a Hochschild cocycle of
the strict zigzag algebra by
TauCeti.hochschildDifferential_obstruction_zigzagLoopAInfinityAlgebra, is not the Hochschild
differential of any suspended cochain of degree zero.
The rotation by one step of the backtrack walk of e.symm and d, for darts d and e with
a common head, is a closed walk: a single coefficient on it is supported on closed walks.
Weights built from single-entry matrices #
Four darts with a common head #
A quartic loop operation which is not a Hochschild coboundary. 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 operation of the closed walk (δ 3).symm, δ 0, (δ 0).symm, δ 3 (in product
order), the rotation of the backtrack walk of (δ 3).symm and δ 0, 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.
The quartic Hochschild cohomology HH^{4, -2} of a zigzag algebra is nonzero when four
distinct darts of G have a common head, that is when some vertex has four distinct neighbours,
over every nontrivial commutative ring: some Hochschild cocycle of the strict zigzag algebra of
suspended degree one, concentrated on words of length four, is not the Hochschild differential of
a cochain of degree zero.
Two branch vertices joined by a walk #
A quartic loop operation at a vertex joined by a walk to another branch vertex which is not a
Hochschild coboundary. 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. Thus v and w each have three distinct neighbours,
as do the two branch vertices of the affine Dₙ graphs for n ≥ 5. The loop operation of the
closed walk e.symm, d, d.symm, e (in product order), the rotation of the backtrack walk of
e.symm and d, 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.
The quartic Hochschild cohomology HH^{4, -2} of a zigzag algebra is nonzero when G has
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, over every nontrivial commutative ring. The hypotheses are those of the
previous theorem, which shows that the loop operation of the walk e.symm, d, d.symm, e is not a
Hochschild coboundary.