The zigzag relation ideal is admissible

Let R be the arrow ideal of the path algebra of the doubled quiver of a finite simple graph G. The uniform zigzag relation ideal is squeezed between R ^ 3 and R ^ 2: every uniform relator is a combination of paths of length at least two, and every path of length at least three is itself a relator. So the zigzag ideal is an admissible ideal, and the zigzag relation quotient is a bound quiver algebra in the sense of TauCeti.IsAdmissibleIdeal, for every finite simple graph.

The lower bound is what makes the image of the arrow ideal cube-zero: whenever the quadratic relators already lie in R ^ 3 (as happens for the one-edge graph A₂, where they all vanish), the zigzag ideal is exactly R ^ 3.

Main results

• TauCeti.arrowIdeal_pow_three_le_zigzagIdeal: R ^ 3 lies in the zigzag ideal.
• TauCeti.zigzagIdeal_le_arrowIdeal_sq: the zigzag ideal lies in R ^ 2.
• TauCeti.isAdmissibleIdeal_zigzagIdeal: the zigzag ideal is admissible.
• TauCeti.zigzagIdeal_eq_arrowIdeal_pow_three: if every quadratic relator lies in R ^ 3, the zigzag ideal is R ^ 3.
• TauCeti.instFiniteDimensionalNonisolatedZigzagQuotient: the zigzag relation quotient is finite-dimensional over a field.

References

See Huerfano--Khovanov, A category for the adjoint representation, Section 3, for the zigzag relations, and Assem--Simson--Skowroński, Elements of the Representation Theory of Associative Algebras I, Ch. II.2, for admissible ideals.

theorem TauCeti.arrowIdeal_pow_three_le_zigzagIdeal (k : Type w) [CommRing k] {V : Type u} (G : SimpleGraph V) [Finite V] : arrowIdeal k (DoubledQuiver G) ^ 3 ≤ TwoSidedIdeal.asIdeal (zigzagIdeal k G)

Every path of length at least three is a zigzag relation: the cube of the arrow ideal of the doubled path algebra lies in the uniform zigzag ideal.

theorem TauCeti.IsZigzagRelator.mem_arrowIdeal_sq (k : Type w) [CommRing k] {V : Type u} (G : SimpleGraph V) [Finite V] {x : pathAlgebra k (DoubledQuiver G)} (hx : IsZigzagRelator k G x) : x ∈ arrowIdeal k (DoubledQuiver G) ^ 2

A uniform zigzag relator lies in the square of the arrow ideal.

theorem TauCeti.zigzagIdeal_le_arrowIdeal_sq (k : Type w) [CommRing k] {V : Type u} (G : SimpleGraph V) [Finite V] : TwoSidedIdeal.asIdeal (zigzagIdeal k G) ≤ arrowIdeal k (DoubledQuiver G) ^ 2

Every zigzag relation has length at least two: the uniform zigzag ideal lies in the square of the arrow ideal of the doubled path algebra.

theorem TauCeti.isAdmissibleIdeal_zigzagIdeal (k : Type w) [CommRing k] {V : Type u} (G : SimpleGraph V) [Finite V] : IsAdmissibleIdeal (TwoSidedIdeal.asIdeal (zigzagIdeal k G))

The zigzag relation ideal is admissible, so the zigzag relation quotient of every finite simple graph is a bound quiver algebra of its doubled quiver.

theorem TauCeti.zigzagIdeal_eq_arrowIdeal_pow_three (k : Type w) [CommRing k] {V : Type u} (G : SimpleGraph V) [Finite V] (h : ∀ (x : pathAlgebra k (DoubledQuiver G)), IsQuadraticZigzagRelator k G x → x ∈ arrowIdeal k (DoubledQuiver G) ^ 3) : TwoSidedIdeal.asIdeal (zigzagIdeal k G) = arrowIdeal k (DoubledQuiver G) ^ 3

The arrow-ideal-cube-zero presentation. When every quadratic zigzag relator is a combination of paths of length at least three, the uniform zigzag ideal is exactly the cube of the arrow ideal.

instance TauCeti.instFiniteDimensionalNonisolatedZigzagQuotient {V : Type u} (G : SimpleGraph V) [Finite V] (K : Type w) [Field K] : FiniteDimensional K (nonisolatedZigzagQuotient K G)

The zigzag relation quotient of every finite simple graph is finite-dimensional, including graphs with isolated vertices, as a bound quiver algebra of the doubled quiver.