The quadratic dual of a zigzag algebra

The zigzag algebra of a finite simple graph G is presented on the doubled quiver by two families of quadratic relations: the length-two paths whose endpoints differ, and the differences of two backtracks based at one vertex. This file computes the orthogonal complement of the span of those relators and identifies the resulting quadratic dual.

The answer is the signless preprojective relation. A degree-two element orthogonal to every length-two path with distinct endpoints is supported on the backtracks, and orthogonality to the differences of the backtracks at a vertex makes its coordinates there constant; so the orthogonal complement is spanned by the sums

s_i = ∑_{j ∼ i} (i → j → i)

of all backtracks at a vertex, one for each vertex, which is the relation Huerfano and Khovanov find in the quadratic dual.

Quadratic duality presents the dual algebra on the opposite quiver, hence — through TauCeti.PathAlgebra.reverseOpAlgEquiv — on the opposite of the doubled path algebra. Every backtrack is a palindrome, so reversal carries the signless relation to itself and the quadratic dual is presented back on the doubled quiver itself: TauCeti.quadraticDualQuadraticZigzagEquivSignless.

That equivalence is a statement about the quadratic presentation, and holds for every finite simple graph. The graph enters only through the identification of the presented algebra with the zigzag algebra, TauCeti.quadraticZigzagPresentationEquivZigzagQuotient, which carries the hypotheses of TauCeti.zigzagIdeal_eq_quadraticZigzagIdeal: connectedness and at least three vertices. It is for such a graph, and only there, that the algebra computed here is the quadratic dual of the zigzag algebra. The one-vertex and two-vertex graphs, whose zigzag algebras are the dual numbers and a radical-cube-zero algebra, are not presented by their quadratic relations and are excluded.

Main definitions

• TauCeti.quadraticZigzagRelations: the k-span of the quadratic zigzag relators, the degree-two relation space of the quadratic presentation.
• TauCeti.quadraticZigzagPresentationEquivZigzagQuotient: for a connected graph with at least three vertices that relation space presents the zigzag algebra.
• TauCeti.quadraticDualQuadraticZigzagEquivSignless: the quadratic dual of that presentation is the signless preprojective algebra of the doubled quiver.

Main results

• TauCeti.twoSidedIdeal_span_quadraticZigzagRelations_eq_quadraticZigzagIdeal and TauCeti.twoSidedIdeal_span_quadraticZigzagRelations_eq_zigzagIdeal: the relation space generates the quadratic zigzag ideal, which for a connected graph with at least three vertices is the zigzag relation ideal, so the quotient it presents is the zigzag algebra.
• TauCeti.quadraticOrthogonal_quadraticZigzagRelations: the orthogonal complement of the quadratic zigzag relations is spanned by the signless relators.

References

S. Huerfano and M. Khovanov, A category for the adjoint representation, Section 6.1, https://arxiv.org/abs/math/0002060, for the quadratic dual of the zigzag algebra.

The coordinates of a signless relator

theorem TauCeti.repr_signlessPreprojectiveRelator_eq_zero (k : Type w) {V : Type u} (G : SimpleGraph V) [Semiring k] [(v : V) → Fintype ↑(G.neighborSet v)] (v : V) {x : Quiver.TotalPath (DoubledQuiver G)} (hx : ∀ (j : V) (h : G.Adj v j), x ≠ ⟨DoubledQuiver.vertex G v, ⟨DoubledQuiver.vertex G v, DoubledQuiver.backtrackPath G h⟩⟩) : ((pathAlgebraBasis k (DoubledQuiver G)).repr (signlessPreprojectiveRelator k (DoubledQuiver.vertex G v))) x = 0

The signless relator at v is supported on the backtracks based at v: its coordinate on any other path vanishes.

theorem TauCeti.repr_signlessPreprojectiveRelator_eq_zero_of_ne (k : Type w) {V : Type u} (G : SimpleGraph V) [Semiring k] [(v : V) → Fintype ↑(G.neighborSet v)] (v : V) {x : Quiver.TotalPath (DoubledQuiver G)} (hx : x.fst ≠ x.snd.fst) : ((pathAlgebraBasis k (DoubledQuiver G)).repr (signlessPreprojectiveRelator k (DoubledQuiver.vertex G v))) x = 0

The coordinate of the signless relator at v on a path which is not a loop vanishes: every backtrack returns to its source.

theorem TauCeti.repr_signlessPreprojectiveRelator_eq_zero_of_length_ne (k : Type w) {V : Type u} (G : SimpleGraph V) [Semiring k] [(v : V) → Fintype ↑(G.neighborSet v)] (v : V) {x : Quiver.TotalPath (DoubledQuiver G)} (hx : x.snd.snd.length ≠ 2) : ((pathAlgebraBasis k (DoubledQuiver G)).repr (signlessPreprojectiveRelator k (DoubledQuiver.vertex G v))) x = 0

The coordinate of the signless relator at v on a path whose length is not two vanishes: every backtrack has length two.

theorem TauCeti.repr_signlessPreprojectiveRelator_backtrackPath_of_ne (k : Type w) {V : Type u} (G : SimpleGraph V) [Semiring k] [(v : V) → Fintype ↑(G.neighborSet v)] (v : V) {i j : V} (h : G.Adj i j) (hvi : v ≠ i) : ((pathAlgebraBasis k (DoubledQuiver G)).repr (signlessPreprojectiveRelator k (DoubledQuiver.vertex G v))) ⟨DoubledQuiver.vertex G i, ⟨DoubledQuiver.vertex G i, DoubledQuiver.backtrackPath G h⟩⟩ = 0

The coordinate of the signless relator at v on a backtrack based at another vertex vanishes.

theorem TauCeti.repr_signlessPreprojectiveRelator_backtrackPath_self (k : Type w) {V : Type u} (G : SimpleGraph V) [Semiring k] [(v : V) → Fintype ↑(G.neighborSet v)] (v : V) {j : V} (h : G.Adj v j) : ((pathAlgebraBasis k (DoubledQuiver G)).repr (signlessPreprojectiveRelator k (DoubledQuiver.vertex G v))) ⟨DoubledQuiver.vertex G v, ⟨DoubledQuiver.vertex G v, DoubledQuiver.backtrackPath G h⟩⟩ = 1

The coordinate of the signless relator at v on a backtrack based at v is one: exactly one summand of the relator is that backtrack.

theorem TauCeti.repr_signlessPreprojectiveRelator_backtrackPath_congr (k : Type w) {V : Type u} (G : SimpleGraph V) [Semiring k] [(v : V) → Fintype ↑(G.neighborSet v)] (v : V) {i j j' : V} (h : G.Adj i j) (h' : G.Adj i j') : ((pathAlgebraBasis k (DoubledQuiver G)).repr (signlessPreprojectiveRelator k (DoubledQuiver.vertex G v))) ⟨DoubledQuiver.vertex G i, ⟨DoubledQuiver.vertex G i, DoubledQuiver.backtrackPath G h⟩⟩ = ((pathAlgebraBasis k (DoubledQuiver G)).repr (signlessPreprojectiveRelator k (DoubledQuiver.vertex G v))) ⟨DoubledQuiver.vertex G i, ⟨DoubledQuiver.vertex G i, DoubledQuiver.backtrackPath G h'⟩⟩

The signless relator has the same coordinate on any two backtracks based at one vertex: this is what makes it orthogonal to the differences of those backtracks.

The quadratic relation space

noncomputable def TauCeti.quadraticZigzagRelations (k : Type w) {V : Type u} (G : SimpleGraph V) [CommRing k] : Submodule k (pathAlgebra k (DoubledQuiver G))

The quadratic relation space of a zigzag algebra: the k-span of the quadratic zigzag relators inside the degree-two part of the path algebra of the doubled quiver.

theorem TauCeti.quadraticZigzagRelations_eq_span (k : Type w) {V : Type u} (G : SimpleGraph V) [CommRing k] : quadraticZigzagRelations k G = Submodule.span k {x : pathAlgebra k (DoubledQuiver G) | IsQuadraticZigzagRelator k G x}

The quadratic relation space is the span of the quadratic relators.

theorem TauCeti.quadraticZigzagRelations_le_grade_two (k : Type w) {V : Type u} (G : SimpleGraph V) [CommRing k] : quadraticZigzagRelations k G ≤ PathAlgebra.grade k (DoubledQuiver G) 2

The quadratic relators are homogeneous of degree two, so the relation space they span lies in the degree-two part of the path algebra.

theorem TauCeti.twoSidedIdeal_span_quadraticZigzagRelations_eq_quadraticZigzagIdeal (k : Type w) {V : Type u} (G : SimpleGraph V) [CommRing k] [Finite V] : TwoSidedIdeal.span ↑(quadraticZigzagRelations k G) = quadraticZigzagIdeal k G

The quadratic relation space generates the quadratic zigzag ideal: passing to the k-span first does not change the two-sided ideal.

theorem TauCeti.twoSidedIdeal_span_quadraticZigzagRelations_eq_zigzagIdeal (k : Type w) {V : Type u} (G : SimpleGraph V) [CommRing k] [Finite V] (hconn : G.Connected) (hcard : 3 ≤ Cardinal.mk V) : TwoSidedIdeal.span ↑(quadraticZigzagRelations k G) = zigzagIdeal k G

For a connected graph with at least three vertices the quadratic relation space presents the zigzag algebra: the two-sided ideal it generates is the zigzag relation ideal.

noncomputable def TauCeti.quadraticZigzagPresentationEquivZigzagQuotient (k : Type w) {V : Type u} (G : SimpleGraph V) [CommRing k] [Finite V] (hconn : G.Connected) (hcard : 3 ≤ Cardinal.mk V) : (pathAlgebra k (DoubledQuiver G) ⧸ TwoSidedIdeal.asIdeal (TwoSidedIdeal.span ↑(quadraticZigzagRelations k G))) ≃ₐ[k] nonisolatedZigzagQuotient k G

For a connected graph with at least three vertices the quadratic relation space presents the zigzag algebra: the quotient of the path algebra of the doubled quiver by the ideal it generates is the zigzag relation quotient. This is what makes the algebra dualised below the zigzag algebra rather than the quadratic presentation alone, and it carries the hypotheses of TauCeti.twoSidedIdeal_span_quadraticZigzagRelations_eq_zigzagIdeal.

@[simp]

theorem TauCeti.quadraticZigzagPresentationEquivZigzagQuotient_mk (k : Type w) {V : Type u} (G : SimpleGraph V) [CommRing k] [Finite V] (hconn : G.Connected) (hcard : 3 ≤ Cardinal.mk V) (x : pathAlgebra k (DoubledQuiver G)) : (quadraticZigzagPresentationEquivZigzagQuotient k G hconn hcard) ((Ideal.Quotient.mk (TwoSidedIdeal.asIdeal (TwoSidedIdeal.span ↑(quadraticZigzagRelations k G)))) x) = (zigzagMk k G) x

The quadratic presentation equivalence sends the class of a path-algebra element to its class in the zigzag quotient.

The orthogonal complement

theorem TauCeti.quadraticOrthogonal_quadraticZigzagRelations (k : Type w) {V : Type u} (G : SimpleGraph V) [CommRing k] [Finite V] [(v : V) → Fintype ↑(G.neighborSet v)] : PathAlgebra.quadraticOrthogonal k (DoubledQuiver G) (quadraticZigzagRelations k G) = Submodule.span k (Set.range fun (v : DoubledQuiver G) => signlessPreprojectiveRelator k v)

The quadratic dual relations of a zigzag algebra are the signless preprojective relations: the orthogonal complement of the quadratic zigzag relations is spanned by the sums of the backtracks at the vertices.

The quadratic dual algebra

theorem TauCeti.signlessPreprojectiveRelator_mem_quadraticOrthogonal (k : Type w) {V : Type u} (G : SimpleGraph V) [CommRing k] [Finite V] [(v : V) → Fintype ↑(G.neighborSet v)] (v : DoubledQuiver G) : signlessPreprojectiveRelator k v ∈ PathAlgebra.quadraticOrthogonal k (DoubledQuiver G) (quadraticZigzagRelations k G)

Every signless relator is a quadratic dual relation of the zigzag algebra.

noncomputable def TauCeti.quadraticDualQuadraticZigzagEquivSignless (k : Type w) {V : Type u} (G : SimpleGraph V) [CommRing k] [Finite V] [(v : V) → Fintype ↑(G.neighborSet v)] : PathAlgebra.quadraticDual k (DoubledQuiver G) (quadraticZigzagRelations k G) ≃ₐ[k] signlessPreprojectiveAlgebra k (DoubledQuiver G)

The quadratic dual of the quadratic presentation of a zigzag algebra is the signless preprojective algebra of the doubled quiver. The quadratic dual is presented on the opposite path algebra; path reversal carries it back to the doubled quiver, fixing every backtrack and hence every signless relator.

The relation space dualised here presents the zigzag algebra for the graphs of TauCeti.quadraticZigzagPresentationEquivZigzagQuotient, connected with at least three vertices, and it is for those graphs that this computes the quadratic dual of the zigzag algebra itself.

@[simp]

theorem TauCeti.quadraticDualQuadraticZigzagEquivSignless_quadraticDualMk (k : Type w) {V : Type u} (G : SimpleGraph V) [CommRing k] [Finite V] [(v : V) → Fintype ↑(G.neighborSet v)] (x : (pathAlgebra k (DoubledQuiver G))ᵐᵒᵖ) : (quadraticDualQuadraticZigzagEquivSignless k G) ((PathAlgebra.quadraticDualMk k (DoubledQuiver G) (quadraticZigzagRelations k G)) x) = (signlessPreprojectiveMk k (DoubledQuiver G)) ((PathAlgebra.reverseOpAlgEquiv k (DoubledQuiver G)).symm x)

The identification of the quadratic dual with the signless algebra is induced by path reversal.

@[simp]

theorem TauCeti.quadraticDualQuadraticZigzagEquivSignless_symm_signlessPreprojectiveMk (k : Type w) {V : Type u} (G : SimpleGraph V) [CommRing k] [Finite V] [(v : V) → Fintype ↑(G.neighborSet v)] (x : pathAlgebra k (DoubledQuiver G)) : (quadraticDualQuadraticZigzagEquivSignless k G).symm ((signlessPreprojectiveMk k (DoubledQuiver G)) x) = (PathAlgebra.quadraticDualMk k (DoubledQuiver G) (quadraticZigzagRelations k G)) ((PathAlgebra.reverseOpAlgEquiv k (DoubledQuiver G)) x)

The inverse identification sends the class of a path-algebra element to the quadratic-dual class obtained by path reversal.