The braid group action of a zigzag algebra on graded modules

Let Z be the zigzag algebra TauCeti.zigzagAlgebra of a finite simple graph G, graded by path length, and let 𝒦 be the homotopy category of unbounded cochain complexes of graded left Z-modules. Tensoring over Z with a complex of graded Z-bimodules is an endofunctor of 𝒦 (CategoryTheory.Functor.map₂HomotopyCategory applied to TauCeti.GradedModuleCat.bimoduleModuleTensor).

For a vertex i, the braid complex Bᵢ = [Pᵢ ⊗_k eᵢ Z ⟶ Z] (TauCeti.zigzagBraidComplex) and its inverse complex Bᵢ' (TauCeti.zigzagBraidInverseComplex) satisfy Bᵢ ⊗_Z Bᵢ' ≃ Z and Bᵢ' ⊗_Z Bᵢ ≃ Z up to homotopy. Hence tensoring with Bᵢ is an autoequivalence σᵢ of 𝒦 whose inverse is tensoring with Bᵢ'. The homotopy equivalences Bᵢ ⊗_Z Bⱼ ≃ Bⱼ ⊗_Z Bᵢ for distinct nonadjacent vertices and Bᵢ ⊗_Z Bⱼ ⊗_Z Bᵢ ≃ Bⱼ ⊗_Z Bᵢ ⊗_Z Bⱼ for adjacent vertices become natural isomorphisms

σᵢ σⱼ ≅ σⱼ σᵢ        and        σᵢ σⱼ σᵢ ≅ σⱼ σᵢ σⱼ

of endofunctors of 𝒦, using the associativity of the action. Therefore the Artin group of G (the braid group of G, generated by symbols σᵢ for the vertices subject to these two families of relations) acts on 𝒦: there is a group homomorphism from it to the group of natural-isomorphism classes of autoequivalences of 𝒦, sending each generator σᵢ to the class of tensoring with Bᵢ. This holds for every finite simple graph and every commutative coefficient ring; no faithfulness of the action is asserted.

Main definitions

• TauCeti.zigzagBraidAutoequivalence: the autoequivalence of 𝒦 given by tensoring with Bᵢ.
• TauCeti.zigzagBraidAutoequivalenceCommIso: the commuting isomorphism for distinct nonadjacent vertices.
• TauCeti.zigzagBraidAutoequivalenceBraidIso: the braid isomorphism for adjacent vertices.
• TauCeti.zigzagBraidArtinAction: the action of the Artin group of G on 𝒦.

References

• M. Khovanov and P. Seidel, Quivers, Floer cohomology, and braid group actions, J. Amer. Math. Soc. 15 (2002), Section 2 and Theorem 2.5, for the type-A action.
• R. S. Huerfano and M. Khovanov, A category for the adjoint representation, J. Algebra 246 (2001), Section 4.6, for the braid group action attached to an arbitrary graph.

noncomputable def TauCeti.zigzagBraidAutoequivalence (k : Type w) [CommRing k] {V : Type u} (G : SimpleGraph V) [Finite V] (i : V) : HomotopyCategory (GradedModuleCat ((zigzagAlgebraInternalGrading k G).piece ℤ)) (ComplexShape.up ℤ) ≌ HomotopyCategory (GradedModuleCat ((zigzagAlgebraInternalGrading k G).piece ℤ)) (ComplexShape.up ℤ)

Tensoring with the braid complex Bᵢ is an autoequivalence of the homotopy category of graded Z-modules, with inverse given by tensoring with the inverse complex Bᵢ'.

@[simp]

theorem TauCeti.zigzagBraidAutoequivalence_functor (k : Type w) [CommRing k] {V : Type u} (G : SimpleGraph V) [Finite V] (i : V) : (zigzagBraidAutoequivalence k G i).functor = (GradedModuleCat.bimoduleModuleTensorHomotopy (zigzagAlgebraInternalGrading k G) (zigzagAlgebraInternalGrading k G)).obj ((HomotopyCategory.quotient (GradedModuleCat (((zigzagAlgebraInternalGrading k G).tensorProduct (zigzagAlgebraInternalGrading k G).opposite).piece ℤ)) (ComplexShape.up ℤ)).obj (zigzagBraidComplex k G i))

The autoequivalence σᵢ is tensoring with the braid complex Bᵢ.

@[simp]

theorem TauCeti.zigzagBraidAutoequivalence_inverse (k : Type w) [CommRing k] {V : Type u} (G : SimpleGraph V) [Finite V] (i : V) : (zigzagBraidAutoequivalence k G i).inverse = (GradedModuleCat.bimoduleModuleTensorHomotopy (zigzagAlgebraInternalGrading k G) (zigzagAlgebraInternalGrading k G)).obj ((HomotopyCategory.quotient (GradedModuleCat (((zigzagAlgebraInternalGrading k G).tensorProduct (zigzagAlgebraInternalGrading k G).opposite).piece ℤ)) (ComplexShape.up ℤ)).obj (zigzagBraidInverseComplex k G i))

The inverse of σᵢ is tensoring with the inverse braid complex Bᵢ'.

noncomputable def TauCeti.zigzagBraidAutoequivalenceCommIso (k : Type w) [CommRing k] {V : Type u} (G : SimpleGraph V) [Finite V] {i j : V} (hij : i ≠ j) (hadj : ¬G.Adj i j) : ((zigzagBraidAutoequivalence k G i).trans (zigzagBraidAutoequivalence k G j)).functor ≅ ((zigzagBraidAutoequivalence k G j).trans (zigzagBraidAutoequivalence k G i)).functor

For distinct nonadjacent vertices the braid autoequivalences commute up to natural isomorphism: applying σᵢ and then σⱼ is isomorphic to applying σⱼ and then σᵢ.

noncomputable def TauCeti.zigzagBraidAutoequivalenceBraidIso (k : Type w) [CommRing k] {V : Type u} (G : SimpleGraph V) [Finite V] {i j : V} (h : G.Adj i j) : ((zigzagBraidAutoequivalence k G i).trans ((zigzagBraidAutoequivalence k G j).trans (zigzagBraidAutoequivalence k G i))).functor ≅ ((zigzagBraidAutoequivalence k G j).trans ((zigzagBraidAutoequivalence k G i).trans (zigzagBraidAutoequivalence k G j))).functor

For adjacent vertices the braid autoequivalences satisfy the braid relation up to natural isomorphism: applying σᵢ, σⱼ, σᵢ in turn is isomorphic to applying σⱼ, σᵢ, σⱼ in turn.

noncomputable def TauCeti.zigzagBraidArtinAction (k : Type w) [CommRing k] {V : Type u} (G : SimpleGraph V) [Finite V] : ArtinGroup G.coxeterMatrix →* AutoequivalenceClass (HomotopyCategory (GradedModuleCat ((zigzagAlgebraInternalGrading k G).piece ℤ)) (ComplexShape.up ℤ))

The braid group action of a zigzag algebra. The Artin group of the graph G acts on the homotopy category of graded Z-modules, by sending the standard generator at a vertex i to the class of tensoring with the braid complex Bᵢ.

@[simp]

theorem TauCeti.zigzagBraidArtinAction_gen (k : Type w) [CommRing k] {V : Type u} (G : SimpleGraph V) [Finite V] (i : V) : (zigzagBraidArtinAction k G) (ArtinGroup.gen G.coxeterMatrix i) = (zigzagBraidAutoequivalence k G i).toAutoequivalenceClass

The standard generator at i acts by tensoring with the braid complex Bᵢ.