Taylor components of composites of graded Taylor expansions

For the reduced tensor coalgebra Tᶜ(M), the arity-n Taylor component of an endomorphism, LinearMap.taylorComponent, is its restriction to words of length n, followed by projection to words of length one.

This file computes the Taylor components of a composite of two graded Taylor expansions. On homogeneous letters, specializing both expansions to the same Taylor map gives the square formula

∑_{r+s+t=n} (-1)^(q (|x₁| + ⋯ + |xᵣ|)) F(x₁,…,xᵣ,F(xᵣ₊₁,…,xᵣ₊ₛ),…,xₙ),

the suspended Stasheff sum when q = 1. The general algebraic fact that a q-twisted coderivation anticommuting with its Koszul twist has an ordinary coderivation as its square is in TauCeti.LinearAlgebra.TensorCoalgebra.GradedCoderivation. It also shows that the square vanishes if and only if every arity component map vanishes.

Anticommutation with the twist expresses oddness when q is odd; when q is even the twist is the identity and the hypothesis reduces to b + b = 0.

Main results

• TauCeti.ReducedTensorWords.taylorComponent_comp_gradedCoderiv_of_tprod: the arity formula for composing an endomorphism with a graded Taylor expansion.
• taylorComponent_comp_gradedCoderiv_of_tprod_of_homogeneous: the corresponding signed formula on homogeneous letters.

References

• E. Getzler and J. D. S. Jones, A-infinity algebras and the cyclic bar complex, Sections 1--2.
• B. Keller, Introduction to A-infinity algebras and modules, Sections 3.1 and 3.6.

theorem TauCeti.ReducedTensorWords.taylorComponent_comp_gradedCoderiv_of_tprod {R : Type uR} {M : Type uM} [CommRing R] [AddCommMonoid M] [Module R M] (G : InternalGrading R M) (b₁ : ReducedTensorWords R M →ₗ[R] ReducedTensorWords R M) (F₂ : ReducedTensorWords R M →ₗ[R] M) (q₂ : ℤ) {n : ℕ} (hn : 0 < n) (x : Fin n → M) : ((b₁ ∘ₗ gradedCoderiv G F₂ q₂).taylorComponent ⟨n, hn⟩) ((PiTensorProduct.tprod R) x) = ∑ p ∈ Finset.range n, ∑ d ∈ Finset.Icc 1 (n - p), (letter R M ∘ₗ b₁) (splice R (G.twistedTuple q₂ x 0 p) 0 n p d (F₂ (subword R x p d)))

The Taylor component of an endomorphism composed with a graded Taylor expansion is obtained by applying the outer endomorphism's letter component to every signed nonempty one-block collapse made by the inner expansion.

theorem TauCeti.ReducedTensorWords.taylorComponent_comp_gradedCoderiv_of_tprod_of_homogeneous {R : Type uR} {M : Type uM} [CommRing R] [AddCommMonoid M] [Module R M] (G : InternalGrading R M) (b₁ : ReducedTensorWords R M →ₗ[R] ReducedTensorWords R M) (F₂ : ReducedTensorWords R M →ₗ[R] M) (q₂ : ℤ) {n : ℕ} (hn : 0 < n) (x : Fin n → M) (𝒟 : Fin n → ℤ) (hx : ∀ (i : Fin n), x i ∈ G.piece (𝒟 i)) : ((b₁ ∘ₗ gradedCoderiv G F₂ q₂).taylorComponent ⟨n, hn⟩) ((PiTensorProduct.tprod R) x) = ∑ p ∈ Finset.range n, ∑ d ∈ Finset.Icc 1 (n - p), ↑↑(q₂ * ∑ j ∈ Finset.range p, if h : j < n then 𝒟 ⟨j, h⟩ else 0).negOnePow • (letter R M ∘ₗ b₁) (splice R x 0 n p d (F₂ (subword R x p d)))

On homogeneous inputs, the Taylor component of such a composite is an insertion sum with the Koszul sign contributed by the inner twist and the degrees of the letters preceding the inserted operation.