Pensieve header: The Drinfel’d-Kohno algebra and associators.

SetDirectory["C:\\drorbn\\AcademicPensieve\\2015-02"];
<< ../Projects/WKO4/FreeLie.m
<< DrinfeldKohno.m

DK (Drinfel’d-Kohno Elements)

b[t[1,2],t[1,3]]

b[t[1,2],t[1,3]]+b[t[1,2],t[2,3]]

b[DK[2,LW[1]], DK[4,LW[2,2,3]+LW[2,3,3]]] // InputForm

σ

t[1,2] // σ[1,34]

t[1,2] // σ[4,3]

DK[3,LW[1,2]] // σ[1,2,34]

DKSeries

DKS[b[t[1,2]+t[2,3], t[3,4]]]

DKS[b[t[1,2]+t[2,3], t[3,4]]] // σ[2,1,3,4]

DKS[b[t[1,2]+t[2,3], t[3,4]]] // σ[1,2,3,45]

DKS[3,αs]

DKS[t[1,2]]

Morphisms LS -> DKS

BCH[DKS[t[1,3]], DKS[t[2,3]]]@{5}

DKS[t[1,2]] // σ[12,3]

DKS[t[1,3]]**DKS[t[2,3]]

Associators

ϕs[2,1]=ϕs[3,1]=ϕs[3,2]=0;ϕs[3,1,2]=-1/24;
{Φ=DKS[3, ϕs], R=DKS[t[1,2]/2]};
HFEqn=(Φ**(Φ//σ[3,2,1]))≡DKS[0];
PentEqn=(Φ**(Φ//σ[1,23,4])**(Φ//σ[2,3,4]))≡((Φ//σ[12,3,4])**(Φ//σ[1,2,34]))

SeriesSolve[Φ, HFEqn&&PentEqn];
Φ@{6}

Φ@{8}

(Φ // σ[1,3,2])@{8}

Φ@{10}

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