Pensieve header: Do solutions of R4, unitarity and cap satisfy the new $\bar{V}^{21}=S_1S_2V^{-1}$?.

SetDirectory["C:\\drorbn\\AcademicPensieve\\Projects\\WKO4"];
<< FreeLie.m;
<< AwCalculus.m;
Rs[a_,b_]:= Es[⟨a->LS[0], b->LS[LW@a]⟩, CWS[0]];
α=LS[{x,y}, αs]; β=LS[{x,y}, βs];γ=CWS[{x,y}, γs];
V=Es[⟨x->α, y->β⟩, γ];
κ=CWS[{x}, κs]; Cap=Es[⟨x->LS[0]⟩,κ];
R4Eqn=V**(Rs[x,z] // dΔ[x,x,y]) ≡ Rs[y,z]**Rs[x,z]**V ;UnitarityEqn=(V**(V//dA[x]//dA[y]) ≡ Es[⟨x->LS[0], y->LS[0]⟩,CWS[0]]);
CapEqn=((V**(Cap//dΔ[x,x,y])//dc[x]//dc[y]) ≡ (Cap*(Cap//dσ[x,y])//dc[x]//dc[y]));
βs[x]= 1/2; βs[y]=0;
SeriesSolve[{α, β,γ, κ}, ((ℏ^(-1))R4Eqn) && UnitarityEqn && CapEqn];
{V, κ}

V@{6} // Timing

Checking ((OverBar[V])^21)=(Subscript[S, 1])(Subscript[S, 2])(V^(-1)).

V//dA[x]//dA[y]

V//dA[x]//dA[y]//dσ[{x,y}->{y,x}]

(V^(-1))

(V^(-1))//dS[x]//dS[y]

(V//dA[x]//dA[y]//dσ[{x,y}->{y,x}])≡((V^(-1))//dS[x]//dS[y])