Vertex-fixing isomorphisms of skew-zigzag algebras and gauge equivalence

A skew-zigzag parameter c of a finite simple graph G labels each ordered pair of incident edges by a unit-valued ratio, and gauge equivalent parameters present isomorphic algebras through an arrow rescaling, which fixes every vertex idempotent. This file proves the converse over a commutative ring: an algebra isomorphism

φ : Z_k(G, c) ≃ₐ[k] Z_k(G, c')

fixing every vertex idempotent forces c and c' to be gauge equivalent. Hence the vertex-fixing isomorphism classes of skew-zigzag algebras of G are exactly the gauge classes of parameters. No grading hypothesis on φ is needed: an isomorphism fixing the idempotents preserves the corner e_j Z e_i between the endpoints of an edge, and that corner is the line spanned by the arrow i ⟶ j. So φ multiplies every arrow by a scalar, necessarily a unit, and applying φ to the defining relation backtrack(h) = c.ratio h h' • backtrack(h') shows that these units gauge c to c'.

Main results

• AlgEquiv.existsUnique_unit_smul_of_vertexFixing: a vertex-fixing isomorphism of skew-zigzag relation quotients multiplies each arrow by a unique unit.
• AlgEquiv.isGaugeEquivalent_of_vertexFixing: a vertex-fixing isomorphism of skew-zigzag relation quotients forces the parameters to be gauge equivalent.
• TauCeti.SkewZigzagParameter.isGaugeEquivalent_iff_exists_vertexFixing_algEquiv: two parameters are gauge equivalent exactly when their relation quotients are isomorphic by a vertex-fixing isomorphism.

References

C. Couture, Skew-Zigzag Algebras, Section 4, https://arxiv.org/abs/1509.08405, which identifies the vertex-fixing graded isomorphism classes of skew-zigzag algebras with the gauge classes of their parameters, and those with the graph cohomology H¹(G, kˣ).

Vertex-fixing isomorphisms

theorem AlgEquiv.existsUnique_unit_smul_of_vertexFixing {k : Type w} {V : Type u} {G : SimpleGraph V} [Finite V] [CommRing k] {c c' : TauCeti.SkewZigzagParameter k G} (φ : TauCeti.skewZigzagQuotient k G c ≃ₐ[k] TauCeti.skewZigzagQuotient k G c') (hφ : ∀ (i : V), φ ((TauCeti.skewZigzagMk k G c) (TauCeti.PathAlgebra.vertexIdempotent k (TauCeti.DoubledQuiver.vertex G i))) = (TauCeti.skewZigzagMk k G c') (TauCeti.PathAlgebra.vertexIdempotent k (TauCeti.DoubledQuiver.vertex G i))) {x y : TauCeti.DoubledQuiver G} (e : x ⟶ y) : ∃! r : kˣ, φ ((TauCeti.skewZigzagMk k G c) (TauCeti.PathAlgebra.ofArrow e)) = ↑r • (TauCeti.skewZigzagMk k G c') (TauCeti.PathAlgebra.ofArrow e)

A vertex-fixing isomorphism rescales arrows by units. An algebra isomorphism between the relation quotients of two parameters that fixes every vertex idempotent multiplies each arrow by a unique unit.

theorem AlgEquiv.isGaugeEquivalent_of_vertexFixing {k : Type w} {V : Type u} {G : SimpleGraph V} [Finite V] [CommRing k] {c c' : TauCeti.SkewZigzagParameter k G} (φ : TauCeti.skewZigzagQuotient k G c ≃ₐ[k] TauCeti.skewZigzagQuotient k G c') (hφ : ∀ (i : V), φ ((TauCeti.skewZigzagMk k G c) (TauCeti.PathAlgebra.vertexIdempotent k (TauCeti.DoubledQuiver.vertex G i))) = (TauCeti.skewZigzagMk k G c') (TauCeti.PathAlgebra.vertexIdempotent k (TauCeti.DoubledQuiver.vertex G i))) : c.IsGaugeEquivalent c'

A vertex-fixing isomorphism forces gauge equivalence. If an algebra isomorphism between the relation quotients of two parameters over a commutative ring fixes every vertex idempotent, then the parameters are gauge equivalent, a gauge being given by the units by which the isomorphism multiplies the arrows.

theorem TauCeti.SkewZigzagParameter.isGaugeEquivalent_iff_exists_vertexFixing_algEquiv {k : Type w} {V : Type u} {G : SimpleGraph V} [Finite V] [CommRing k] {c c' : SkewZigzagParameter k G} : c.IsGaugeEquivalent c' ↔ ∃ (φ : skewZigzagQuotient k G c ≃ₐ[k] skewZigzagQuotient k G c'), ∀ (i : V), φ ((skewZigzagMk k G c) (PathAlgebra.vertexIdempotent k (DoubledQuiver.vertex G i))) = (skewZigzagMk k G c') (PathAlgebra.vertexIdempotent k (DoubledQuiver.vertex G i))

Gauge classes are vertex-fixing isomorphism classes. Two skew-zigzag parameters over a commutative ring are gauge equivalent exactly when their relation quotients are isomorphic by an algebra isomorphism fixing every vertex idempotent.