Pensieve Header: The playground - a minimalistic Lie quotient of ${\mathcal A}^w$, with a BCH product law.

SetDirectory["C:\\drorbn\\AcademicPensieve\\Projects\\w-Computations"];
<< "Playground Program.m"

{ar[1,2]**ar[1,3], ar[1,3]**ar[1,2]}

ToDegree[4, R[1,3]**R[2,3]] // PCollect

ToDegree[4, R[1,2]**R[1,3]**R[2,3]] // PCollect

ToDegree[4, R[1,2]**R[1,3]**R[2,3] - R[2,3]**R[1,3]**R[1,2]]

R[1,2]~Up~{{1,2},3}

fb =-(1/4)+h ((x/32)+((5 y)/96))+(((h^3) (-57 (x^3)-169 (x^2) y-187 x (y^2)-99 (y^3)))/138240)+(1/69672960)(h^5) (555 (x^5)+2558 (x^4) y+5000 (x^3) (y^2)+5078 (x^2) (y^3)+2669 x (y^4)+780 (y^5))+(1/200658124800)(h^7) (-31989 (x^7)-205357 (x^6) y-592097 (x^5) (y^2)-964985 (x^4) (y^3)-949967 (x^3) (y^4)-565943 (x^2) (y^5)-186139 x (y^6)-34275 (y^7));
Vb=PCollect[
(fb /. {x->h[2], y->h[1]})h*ar[2,1] + (1/2 + fb /. {x->h[1], y->h[2]})h*ar[1,2]
+ 0 h (ar[1,1]+  ar[2,2])
+ 1/96 h^2 (h[1] ar[2,2]+h[2]ar[1,1])+0 h^2 (h[1]ar[1,1]+h[2]ar[2,2])
]

ToDegree[2, Vb]

ToDegree[4,Vb**(R[1,3]+R[2,3])]

ToDegree[4, {R4Eqn[Vb], TwistEqn[Vb], TwistMinusEqn[Vb]}]

ToDegree[2, Phi[Vb]]

Phi[4, Vb]

ToDegree[2, #[Phi[Vb]]& /@ {Pentagon, Hexagon, uUnitarity, HexagonMinus}]

ToDegree[4, #[Phi[Vb]]& /@ {Pentagon, Hexagon, uUnitarity, HexagonMinus}]

S1[Phi_]:={
Phi~Up~{1,3,{2,4}},
(-Phi)~Up~{3,2,4},
RR[3,2],
Phi~Up~{2,3,4},
(-Phi)~Up~{1,2,{3,4}}
};
ToDegree[1, S1[Phi[Vb]]]

S2[Phi_] := PCollect[NonCommutativeMultiply @@ S1[Phi]];
ToDegree[2, S2[Phi[Vb]]]

S3[V_] := PCollect /@ {
(-V)~Up~{{1,3},{2,4}},
(-V)~Up~{1,3},
(-V)~Up~{2,4},
R[3,2],
V,
V~Up~{3,4},
V~Up~{{1,2},{3,4}}
};
ToDegree[1, S3[Vb]]

S4[V_] := PCollect[NonCommutativeMultiply @@ S3[V]];
ToDegree[2, S4[Vb]]

ToDegree[4, S2[Phi[Vb]]-S4[Vb]]  // PCollect

ToDegree[8, Puncture[2, -Vb]]

CoefficientList[ToDegree[8, Puncture[2, Vb]] /. {h[1] -> 1, ar[1,2]->1}, h]

Series[Log[(E^(x/2)-E^(-x/2))/x], {x, 0, 8}]

Series[Log[(E^x-1)/x], {x, 0, 8}]

( -(1/4)+((x/32)+((5 y)/96)) h+(((-57 (x^3)-169 (x^2) y-187 x (y^2)-99 (y^3)) (h^3))/138240)+(1/69672960)(555 (x^5)+2558 (x^4) y+5000 (x^3) (y^2)+5078 (x^2) (y^3)+2669 x (y^4)+780 (y^5)) (h^5)+(1/200658124800)(-31989 (x^7)-205357 (x^6) y-592097 (x^5) (y^2)-964985 (x^4) (y^3)-949967 (x^3) (y^4)-565943 (x^2) (y^5)-186139 x (y^6)-34275 (y^7)) (h^7)+(1/1589212348416000)(5104911 (x^9)+42750073 (x^8) y+165124408 (x^7) (y^2)+376113312 (x^6) (y^3)+552802446 (x^5) (y^4)+542218746 (x^4) (y^5)+352835712 (x^3) (y^6)+146191048 (x^2) (y^7)+33851563 x (y^8)+4104741 (y^9)) (h^9)+((O[h])^11)) /. y->0