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TauCeti.RepresentationTheory.Quiver.Preprojective.Signless.Affine.D

Nonzero cyclic classes from affine Dₙ subgraphs #

Let G be a finite simple graph containing a copy of the affine diagram Dₙ (n ≥ 4), not necessarily induced. This file shows that the cyclic quotient Π⁺(G) / [Π⁺(G), Π⁺(G)] of the signless preprojective algebra of G, its zeroth Hochschild homology, is nonzero in path degree four, over every nontrivial commutative ring. For a bipartite graph, such as a tree, Π⁺(G) is isomorphic to the additive preprojective algebra of any orientation of G by a rescaling of the arrows by signs (SimpleGraph.IsBipartite.signlessPreprojectiveAlgebraEquiv).

For a quiver Q whose underlying graph is a tree (bipartite, so Π⁺ is the additive Π(Q)), Liu and Wang identify the degree-(q + 2) piece of this cyclic quotient with the second Hochschild cohomology HH^{2,q} of the two-dimensional Ginzburg algebra Π₂(Q), which controls the A∞ deformations of the zigzag algebra of the tree. For the preprojective algebra of a Dynkin quiver over a field of characteristic zero, the cyclic quotient is concentrated in degree zero (Etingof--Eu). The present computation exhibits a positive-degree class for every graph containing an affine Dₙ, for instance for every tree with a vertex of degree at least four or with two vertices of degree at least three.

The representation #

Label Dₙ as in TauCeti.AffineDynkinType.graph: the path 0 - 1 - ⋯ - (n - 2), with a leaf n - 1 at node 1 and a leaf n at node n - 3. Put k² at every node and assign the following matrices to the arrows, writing Eᵢⱼ for the matrix units, J = E₂₁ - E₁₂ and D = E₁₁ - E₂₂:

0 → 1 : E₁₁      1 → 0 : E₁₂          (n-1) → 1 : E₂₂     1 → (n-1) : E₂₁
i → i+1 : J      i+1 → i : D          for 1 ≤ i < n - 3
(n-2) → (n-3) : -E₁₁   (n-3) → (n-2) : E₁₂    n → (n-3) : -E₂₂   (n-3) → n : E₂₁

At node 1 the two leaves contribute the backtracks E₁₂ and E₂₁, whose sum is the swap matrix N = E₁₂ + E₂₁. Along each path edge the backtracks are D J = -N at its lower end and J D = N at its upper end, so the relations hold along the path, and at node n - 3 the two remaining leaves contribute -E₁₂ - E₂₁ = -N. Every backtrack at a leaf is zero. The resulting representation of Π⁺(Dₙ) extends by zero to Π⁺(G) along the copy.

The walk 1 → 0 → 1 → (n - 1) → 1 at the branch node, the product of the backtrack through the leaf n - 1 and the backtrack through the leaf 0, acts at node 1 by E₂₁ E₁₂ = E₂₂, of trace one. Since the trace of a representation vanishes on commutators, the class of this walk in the cyclic quotient is nonzero.

Main definitions #

Main results #

References #

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noncomputable def TauCeti.affineDMatrix (k : Type w) [CommRing k] (n i j : ℕ) :
Matrix (Fin 2) (Fin 2) k

The matrix assigned to the arrow i → j of the doubled affine diagram Dₙ by the representation of this file, with nodes given as natural numbers in the labelling of TauCeti.AffineDynkinType.graph; it is zero unless i and j are adjacent.

Instances For

    Evaluating the matrices #

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    @[simp]
    theorem TauCeti.affineDMatrix_eq_zero_of_not_adj (k : Type w) [CommRing k] {n : ℕ} (hn : 4 ≤ n) {i j : Fin (AffineDynkinType.D n).nodes} (h : ¬(AffineDynkinType.D n).graph.Adj i j) :
    affineDMatrix k n ↑i ↑j = 0

    The affine Dₙ matrices vanish off the edges of the diagram.

    source
    theorem TauCeti.sum_affineDMatrix_mul_affineDMatrix (k : Type w) [CommRing k] {n : ℕ} (hn : 4 ≤ n) (v : Fin (AffineDynkinType.D n).nodes) :
    ∑ w ∈ (AffineDynkinType.D n).graph.neighborFinset v, affineDMatrix k n ↑w ↑v * affineDMatrix k n ↑v ↑w = 0

    The affine Dₙ matrices satisfy the signless relations: at every node v of Dₙ, the backtracks β w v * β v w over the neighbours w of v sum to zero.

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    theorem TauCeti.trace_affineDMatrix_backtracks (k : Type w) [CommRing k] {n : ℕ} (hn : 4 ≤ n) :
    (affineDMatrix k n (n - 1) 1 * affineDMatrix k n 1 (n - 1) * (affineDMatrix k n 0 1 * affineDMatrix k n 1 0)).trace = 1

    In the affine Dₙ representation, the product of the backtracks at the branch node 1 through the leaf n - 1 and then through the leaf 0 has trace one.

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    theorem TauCeti.exists_mem_signlessPreprojectiveGrade_four_notMem_commutatorSubmodule {k : Type w} [CommRing k] {V : Type u} [Fintype V] {G : SimpleGraph V} [DecidableRel G.Adj] [Nontrivial k] {n : ℕ} (hn : 4 ≤ n) (hG : (AffineDynkinType.D n).graph.IsContained G) :
    ∃ x ∈ signlessPreprojectiveGrade k (DoubledQuiver G) 4, x ∉ commutatorSubmodule k (signlessPreprojectiveAlgebra k (DoubledQuiver G))

    Graphs containing an affine Dₙ have nonzero cyclic classes in degree four. If G contains a copy of the affine diagram Dₙ, not necessarily induced, then over every nontrivial commutative ring some element of path degree four of the signless preprojective algebra Π⁺(G) does not lie in the commutator submodule [Π⁺(G), Π⁺(G)]: its class in the cyclic quotient is nonzero.

    The element is the closed walk 1 → 0 → 1 → (n - 1) → 1 at the image of the branch node 1; its image under the representation of this file has trace one.