The cancellable blocks in an adjacent braid product #
Evaluating the middle factor of the degree-two arrow insertion gives the volume insertion in the self-tensor. This identifies the invertible component of the lowest differential of the actual adjacent triple product.
References #
- R. S. Huerfano and M. Khovanov, A category for the adjoint representation, arXiv:math/0002060, Sections 3 and 4.6, Proposition 7.
- M. Khovanov and P. Seidel, Quivers, Floer cohomology, and braid group actions, arXiv:math/0006056, Theorem 2.5 for the type-A relation.
@[simp]
theorem
TauCeti.zigzagBraidAdjacentTripleInsert_evaluateMiddle
(k : Type u_1)
[CommRing k]
{V : Type u_2}
(G : SimpleGraph V)
[Finite V]
{i j : V}
(h : G.Adj j i)
:
CategoryTheory.CategoryStruct.comp (zigzagBraidAdjacentTripleInsert k G h)
(GradedModuleCat.bimoduleTensorMap (zigzagAlgebraInternalGrading k G) (zigzagAlgebraInternalGrading k G)
(zigzagAlgebraInternalGrading k G) (zigzagBraidTensorEvaluationRight k G i j)
(CategoryTheory.CategoryStruct.id (zigzagGradedBraidBimodule k G i))) = zigzagBraidSelfTensorInr k G i
Evaluating the middle vertex of the adjacent top tensor inserts the volume in the self-tensor, with no internal shift lost.
@[simp]
theorem
TauCeti.zigzagBraidAdjacentTripleInsert_evaluateMiddle_assoc
(k : Type u_1)
[CommRing k]
{V : Type u_2}
(G : SimpleGraph V)
[Finite V]
{i j : V}
(h : G.Adj j i)
{Z : GradedModuleCat ((zigzagEnvelopingGrading k G).piece ℤ)}
(h✝ :
GradedModuleCat.bimoduleTensorObj (zigzagAlgebraInternalGrading k G) (zigzagAlgebraInternalGrading k G)
(zigzagAlgebraInternalGrading k G) (zigzagGradedBraidBimodule k G i) (zigzagGradedBraidBimodule k G i) ⟶ Z)
:
CategoryTheory.CategoryStruct.comp (zigzagBraidAdjacentTripleInsert k G h)
(CategoryTheory.CategoryStruct.comp
(GradedModuleCat.bimoduleTensorMap (zigzagAlgebraInternalGrading k G) (zigzagAlgebraInternalGrading k G)
(zigzagAlgebraInternalGrading k G) (zigzagBraidTensorEvaluationRight k G i j)
(CategoryTheory.CategoryStruct.id (zigzagGradedBraidBimodule k G i)))
h✝) = CategoryTheory.CategoryStruct.comp (zigzagBraidSelfTensorInr k G i) h✝
Evaluating the middle vertex of the adjacent top tensor inserts the volume in the self-tensor, with no internal shift lost.