Pensieve header: Finding $Z^w$ starting from a non-even associator, to test if cap is always $\nu^{1/4}$. Continues WKOSession.nb of pensieve://Projects/WKO4/.

SetDirectory["C:\\drorbn\\AcademicPensieve\\Projects\\Duflo"];
<< ../WKO4/FreeLie.m;
<< ../WKO4/AwCalculus.m;

DK2Es[s___][ζ_] := El[ζ // αMap[s], CWS[0]] // Γ;

Φs[2,1]=Φs[3,1]=Φs[3,2]=0;Φs[3,1,2]=1/24; Φs[3,1,1,2]=1/7; Φs[3,1,1,1,1,2]=1/13;
(Subscript[Φ, 0])=DKS[3, Φs];
SeriesSolve[(Subscript[Φ, 0]), (((Subscript[Φ, 0])^(σ[3,2,1]))≡-(Subscript[Φ, 0]))∧((Subscript[Φ, 0])**((Subscript[Φ, 0])^(σ[1,23,4]))**((Subscript[Φ, 0])^(σ[2,3,4]))≡((Subscript[Φ, 0])^(σ[12,3,4]))**((Subscript[Φ, 0])^(σ[1,2,34])))];
(Subscript[Φ, 0])@{6}

R = DKS[t[1,2]/2];
(Subscript[Z, B]) = (((-(Subscript[Φ, 0])))^(σ[13,2,4]))**((Subscript[Φ, 0])^(σ[1,3,2]))**(R^(σ[2,3]))**(((-(Subscript[Φ, 0])))^(σ[1,2,3]))**((Subscript[Φ, 0])^(σ[12,3,4]))

(Subscript[V, B]) = (Subscript[Z, B]) // DK2Es[1,2,3,4] // ((tη)^1) // ((tη)^3) // ((hη)^2) // ((hη)^4) // hσ[{1,3}->{x,y}] // tσ[{2,4}->{x,y}]

γ=CWS[{x,y}, γs];
(Subscript[V, 0])=Es[(Subscript[V, B])[[1]], γ];

κ=CWS[{x}, κs]; Cap=Es[⟨x->LS[0]⟩,κ];

Rs[a_,b_]:= Es[⟨a->LS[0], b->LS[LW@a]⟩, CWS[0]];
R4Eqn=(Subscript[V, 0])**(Rs[x,z] // dΔ[x,x,y]) ≡ Rs[y,z]**Rs[x,z]**(Subscript[V, 0]) ;UnitarityEqn=(Subscript[V, 0])**((Subscript[V, 0])//dA) ≡ Es[⟨x->LS[0], y->LS[0]⟩,CWS[0]];
CapEqn=((Subscript[V, 0])**(Cap//dΔ[x,x,y])//dc[x]//dc[y]) ≡ (Cap*(Cap//dσ[x,y])//dc[x]//dc[y]);

SeriesSolve[{γ, κ}, ((ℏ^(-1))R4Eqn) ∧ UnitarityEqn ∧ CapEqn];
{(Subscript[V, 0])@{4}, κ@{6}}

Θl[x_,y_, s_] := El[⟨x->LS[s LW@y], y->LS[s LW@x]⟩, CWS[0]];
Θs[x_,y_,s_] := Θl[x,y,s] // Γ;
{Θl[x,y,1], Θs[x,y,1]}

τV=Rs[x,y]**((Subscript[V, 0])//dσ[{x,y}->{y,x}])**Θs[x,y,-1/2];
((Subscript[V, 0])≡τV)@{6}