Graded isomorphism classes of skew-zigzag algebras

The skew-zigzag relation quotients of two parameters on a finite simple graph are graded isomorphic exactly when their gauge classes differ by a graph automorphism. Equivalently, their classes in H¹(G, kˣ) lie in the same orbit of Aut(G).

The coefficient ring is any nontrivial commutative ring with only trivial idempotents. In particular, every field qualifies, with IsIdempotentElem.iff_eq_zero_or_one supplying this condition; neither square roots nor a restriction on the characteristic are needed. The graph need not be connected. These statements concern skewZigzagQuotient, the relation quotient, whose isolated-vertex factors are copies of k; they do not replace those factors with the dual numbers used in the public ordinary zigzagAlgebra.

Gradedness is expressed by preservation and reflection of membership in every path-length piece, as in the vertex-fixing classification in TauCeti.RepresentationTheory.Quiver.Zigzag.Skew.Grading. No condition on the images of the vertex idempotents is imposed: a graded isomorphism necessarily permutes them. Conversely, relabelling followed by a gauge isomorphism supplies a graded isomorphism.

Main results

• TauCeti.SkewZigzagParameter.exists_isGaugeEquivalent_relabel_iff_exists_graded_algEquiv: the gauge criterion for arbitrary graded isomorphisms.
• TauCeti.SkewZigzagParameter.exists_firstCohomologyRelabel_eq_iff_exists_graded_algEquiv: the cohomological classification by graph-automorphism orbits.

References

C. Couture, Skew-Zigzag Algebras, Section 4, Theorem 4.12, https://arxiv.org/abs/1509.08405, proves the graded classification for connected graphs over fields containing square roots. Here the existing gauge and vertex-permutation criteria give the classification under the weaker coefficient and graph hypotheses above.

theorem TauCeti.SkewZigzagParameter.exists_isGaugeEquivalent_relabel_iff_exists_graded_algEquiv {k : Type w} [CommRing k] [Nontrivial k] {V : Type u} [Finite V] {G : SimpleGraph V} {c c' : SkewZigzagParameter k G} (hidempotents : ∀ (e : k), IsIdempotentElem e → e = 0 ∨ e = 1) : (∃ (τ : G ≃g G), (relabel τ c).IsGaugeEquivalent c') ↔ ∃ (φ : skewZigzagQuotient k G c ≃ₐ[k] skewZigzagQuotient k G c'), ∀ (n : ℕ) (x : skewZigzagQuotient k G c), φ x ∈ skewZigzagGrade k G c' n ↔ x ∈ skewZigzagGrade k G c n

Gauge classes modulo graph automorphisms are the graded isomorphism classes. Over a nontrivial commutative ring with only trivial idempotents, two skew-zigzag relation quotients are graded isomorphic exactly when one parameter becomes gauge equivalent to the other after relabelling by a graph automorphism. The isomorphism is not required to fix the vertices.

theorem TauCeti.SkewZigzagParameter.exists_firstCohomologyRelabel_eq_iff_exists_graded_algEquiv {k : Type w} [CommRing k] [Nontrivial k] {V : Type u} [Finite V] {G : SimpleGraph V} {c c' : SkewZigzagParameter k G} (hidempotents : ∀ (e : k), IsIdempotentElem e → e = 0 ∨ e = 1) : (∃ (τ : G ≃g G), (SimpleGraph.firstCohomologyRelabel kˣ τ) ((cohomologyClass k G) c) = (cohomologyClass k G) c') ↔ ∃ (φ : skewZigzagQuotient k G c ≃ₐ[k] skewZigzagQuotient k G c'), ∀ (n : ℕ) (x : skewZigzagQuotient k G c), φ x ∈ skewZigzagGrade k G c' n ↔ x ∈ skewZigzagGrade k G c n

Couture's graded classification by Aut(G)-orbits on H¹(G, kˣ). Over a nontrivial commutative ring with only trivial idempotents, the cohomology classes of two parameters lie in the same graph-automorphism orbit exactly when their skew-zigzag relation quotients are graded isomorphic. Together with cohomologyClass_surjective, this describes all graded isomorphism classes of the relation quotients arising from skew-zigzag parameters.