Class bases and the graded Cartan map of the componentwise zigzag algebra

The graded vertex projectives of the public zigzag algebra form a basis of its actual Laurent Grothendieck group of finite graded projectives. This includes isolated vertices, whose projectives retain the degree-two generator of the dual-number factor. These are the projectives on which the zigzag braid complexes act, rather than projectives of the relation quotient with its different isolated-vertex convention.

The graded Cartan map sends a vertex-projective class to a class with coordinate 1 + q² at that vertex, q at adjacent vertices, and zero elsewhere. The heads of the vertex projectives form the simple-class basis of the graded module Grothendieck group, with the idempotent coordinates as its coordinates. Thus the matrix of the categorical graded Cartan map in these two bases is (1 + q²) I + q A_G, including its dual-number diagonal entries at isolated vertices.

References

• Huerfano--Khovanov, A category for the adjoint representation, Sections 3 and 4.6.
• Ehrig--Tubbenhauer, Algebraic properties of zigzag algebras, Section 2.
• The graded Krull–Schmidt and simple-head basis constructions are TauCeti.gradedIndecomposableProjectiveClassBasis and TauCeti.gradedIdempotentHeadBasis.

theorem TauCeti.finrank_zigzagAlgebraGradedProjective_hom (k : Type w) [Field k] {V : Type u} [Finite V] (G : SimpleGraph V) (i j : V) : Module.finrank k (zigzagAlgebraGradedProjective k G i ⟶ zigzagAlgebraGradedProjective k G j) = if i = j then 1 else 0

Degree-preserving maps between vertex projectives are scalar endomorphisms at the same vertex and vanish between distinct vertices.

theorem TauCeti.indecomposable_zigzagAlgebraGradedProjective (k : Type w) [Field k] {V : Type u} [Finite V] (G : SimpleGraph V) (i : V) : CategoryTheory.Indecomposable { obj := zigzagAlgebraGradedProjective k G i, property := ⋯ }

Every vertex projective of the public algebra is indecomposable as a finite graded projective, including the dual-number projective at an isolated vertex.

theorem TauCeti.isEmpty_iso_zigzagAlgebraGradedProjective_shiftObj (k : Type w) [Field k] {V : Type u} [Finite V] (G : SimpleGraph V) {i j : V} (hij : i ≠ j) (d : ℤ) : IsEmpty (zigzagAlgebraGradedProjective k G i ≅ (zigzagAlgebraGradedProjective k G j).shiftObj d)

Distinct vertex projectives of the public algebra cannot become isomorphic after an internal shift.

theorem TauCeti.isExhaustiveGradedIndecomposableProjectiveFamily_zigzagAlgebraGradedProjective (k : Type w) [Field k] {V : Type u} [Finite V] (G : SimpleGraph V) : IsExhaustiveGradedIndecomposableProjectiveFamily fun (i : V) => { obj := zigzagAlgebraGradedProjective k G i, property := ⋯ }

Every finite graded indecomposable projective of the public zigzag algebra is an internal shift of a vertex projective.

noncomputable def TauCeti.zigzagAlgebraGradedProjectiveClassBasis (k : Type w) [Field k] {V : Type u} [Finite V] (G : SimpleGraph V) : Module.Basis V (LaurentPolynomial ℤ) (LaurentK0 (gradedFiniteProjectiveModulesExactStructure (zigzagAlgebraIntegerGrade k G)))

The vertex-projective class basis of K₀^gr(proj Z) for the public componentwise algebra. Its Laurent scalar action represents internal grading shifts.

@[simp]

theorem TauCeti.zigzagAlgebraGradedProjectiveClassBasis_apply (k : Type w) [Field k] {V : Type u} [Finite V] (G : SimpleGraph V) (i : V) : (zigzagAlgebraGradedProjectiveClassBasis k G) i = zigzagAlgebraGradedProjectiveClass k G i

The basis vector at a vertex is the class of its actual graded vertex projective.

@[simp]

theorem TauCeti.zigzagAlgebraGradedProjectiveClassBasis_repr_class (k : Type w) [Field k] {V : Type u} [Finite V] (G : SimpleGraph V) (i : V) : (zigzagAlgebraGradedProjectiveClassBasis k G).repr (zigzagAlgebraGradedProjectiveClass k G i) = Finsupp.single i 1

The coordinate of a vertex-projective class is the corresponding coordinate vector.

The simple-class basis and the categorical Cartan matrix

noncomputable def TauCeti.zigzagAlgebraGradedSimple (k : Type w) [Field k] {V : Type u} [Finite V] (G : SimpleGraph V) (i : V) : GradedModuleCat (zigzagAlgebraIntegerGrade k G)

The graded head of the public vertex projective, obtained by killing its positive-degree part. At an isolated vertex this kills the degree-two dual-number generator.

theorem TauCeti.zigzagAlgebraGradedSimple_def (k : Type w) [Field k] {V : Type u} [Finite V] (G : SimpleGraph V) (i : V) : zigzagAlgebraGradedSimple k G i = have x := zigzagAlgebraIntegerGradedAlgebra k G; gradedPositiveMulQuotient (zigzagAlgebraIntegerGrade k G) ⋯

A graded vertex simple is the generic graded head of its vertex idempotent.

noncomputable def TauCeti.zigzagAlgebraGradedSimpleMk (k : Type w) [Field k] {V : Type u} [Finite V] (G : SimpleGraph V) (i : V) : ↥(Ideal.span {(fun (i : V) => (zigzagAlgebraBasis k G) (Sum.inl i)) i}) →ₗ[↑(zigzagAlgebra k G)] (zigzagAlgebraGradedSimple k G i).carrier

The class of a vertex-projective element in its graded simple head.

theorem TauCeti.zigzagAlgebraGradedSimpleMk_surjective (k : Type w) [Field k] {V : Type u} [Finite V] (G : SimpleGraph V) (i : V) : Function.Surjective ⇑(zigzagAlgebraGradedSimpleMk k G i)

Every element of the graded simple head has a vertex-projective representative.

@[simp]

theorem TauCeti.zigzagAlgebraGradedSimpleMk_eq_zero_iff (k : Type w) [Field k] {V : Type u} [Finite V] (G : SimpleGraph V) (i : V) (x : ↥(Ideal.span {(fun (i : V) => (zigzagAlgebraBasis k G) (Sum.inl i)) i})) : (zigzagAlgebraGradedSimpleMk k G i) x = 0 ↔ ↑x ∈ gradedPositiveMulIdeal (zigzagAlgebraIntegerGrade k G) ((fun (i : V) => (zigzagAlgebraBasis k G) (Sum.inl i)) i)

A representative vanishes in the graded head exactly when it lies in the positive-degree ideal of its vertex projective.

theorem TauCeti.zigzagAlgebraGradedSimpleMk_mem_piece (k : Type w) [Field k] {V : Type u} [Finite V] (G : SimpleGraph V) (i : V) {p : ℤ} (x : ↥(Ideal.span {(fun (i : V) => (zigzagAlgebraBasis k G) (Sum.inl i)) i})) (hx : ↑x ∈ zigzagAlgebraIntegerGrade k G p) : (zigzagAlgebraGradedSimpleMk k G i) x ∈ (zigzagAlgebraGradedSimple k G i).grading.piece ℤ p

The head projection sends a homogeneous representative of degree p to the degree-p piece of the graded vertex simple.

theorem TauCeti.gradedFiniteModules_zigzagAlgebraGradedSimple (k : Type w) [Field k] {V : Type u} [Finite V] (G : SimpleGraph V) (i : V) : gradedFiniteModules (zigzagAlgebraIntegerGrade k G) (zigzagAlgebraGradedSimple k G i)

The head of a public vertex projective is a finite graded module.

theorem TauCeti.simple_zigzagAlgebraGradedSimple (k : Type w) [Field k] {V : Type u} [Finite V] (G : SimpleGraph V) (i : V) : CategoryTheory.Simple (zigzagAlgebraGradedSimple k G i)

Each graded vertex head of the public algebra is simple.

instance TauCeti.instFiniteCarrierCarrierZigzagAlgebraZigzagAlgebraGradedSimple (k : Type w) [Field k] {V : Type u} [Finite V] (G : SimpleGraph V) (i : V) : Module.Finite k (zigzagAlgebraGradedSimple k G i).carrier

A graded vertex simple is finite-dimensional.

@[simp]

theorem TauCeti.smulGradedDimension_zigzagAlgebraGradedSimple (k : Type w) [Field k] {V : Type u} [Finite V] (G : SimpleGraph V) [DecidableEq V] (j i : V) : GradedModuleCat.smulGradedDimension ((fun (i : V) => (zigzagAlgebraBasis k G) (Sum.inl i)) j) (zigzagAlgebraGradedSimple k G i) = if j = i then 1 else 0

The graded vertex simple is concentrated in degree zero, with dimension one at its own vertex and zero at every other vertex.

noncomputable def TauCeti.zigzagAlgebraGradedSimpleClassBasis (k : Type w) [Field k] {V : Type u} [Finite V] (G : SimpleGraph V) : Module.Basis V (LaurentPolynomial ℤ) (LaurentK0 (gradedFiniteModulesExactStructure (zigzagAlgebraIntegerGrade k G)))

The vertex-head class basis of G₀^gr(mod Z) for the public componentwise zigzag algebra.

@[simp]

theorem TauCeti.zigzagAlgebraGradedSimpleClassBasis_apply (k : Type w) [Field k] {V : Type u} [Finite V] (G : SimpleGraph V) (i : V) : (zigzagAlgebraGradedSimpleClassBasis k G) i = LaurentK0.of (gradedFiniteModulesExactStructure (zigzagAlgebraIntegerGrade k G)) { obj := zigzagAlgebraGradedSimple k G i, property := ⋯ }

The simple-class basis vector is the class of the graded head at that vertex.

@[simp]

theorem TauCeti.zigzagAlgebraGradedSimpleClassBasis_repr_apply (k : Type w) [Field k] {V : Type u} [Finite V] (G : SimpleGraph V) (x : LaurentK0 (gradedFiniteModulesExactStructure (zigzagAlgebraIntegerGrade k G))) (i : V) : ((zigzagAlgebraGradedSimpleClassBasis k G).repr x) i = (gradedIdempotentCoordinate ⋯ ⋯) x

The simple-class coordinates are the graded dimensions of the vertex-idempotent pieces.

theorem TauCeti.zigzagAlgebraGradedSimpleClassBasis_repr_gradedCartanMap (k : Type w) [Field k] {V : Type u} [Finite V] (G : SimpleGraph V) (i j : V) : ((zigzagAlgebraGradedSimpleClassBasis k G).repr ((gradedCartanMap (zigzagAlgebraIntegerGrade k G)) (zigzagAlgebraGradedProjectiveClass k G j))) i = if i = j then 1 + LaurentPolynomial.T 2 else if G.Adj i j then LaurentPolynomial.T 1 else 0

Entrywise, the categorical Cartan map records 1 + q² at the projective's vertex, q at an adjacent vertex, and zero otherwise.

theorem TauCeti.gradedCartanMatrix_zigzagAlgebra (k : Type w) [Field k] {V : Type u} [Finite V] (G : SimpleGraph V) [Fintype V] [DecidableEq V] [DecidableRel G.Adj] : gradedCartanMatrix (zigzagAlgebraIntegerGrade k G) (zigzagAlgebraGradedProjectiveClassBasis k G) (zigzagAlgebraGradedSimpleClassBasis k G) = (1 + LaurentPolynomial.T 2) • 1 + LaurentPolynomial.T 1 • SimpleGraph.adjMatrix (LaurentPolynomial ℤ) G

The categorical graded Cartan map of the public algebra has matrix (1 + q²) I + q A_G in the vertex-projective and vertex-head class bases, including isolated vertices.