Pensieve Header: The programs running the playground - a minimalistic Lie quotient of A^w, with a BCH product law.

The cell below is copied from BCH.nb:

BCH[b_, x_, y_] := Module[
{BCH8,B, X, Y, Ev},
BCH8 = B[X,Y]/2+B[X,B[X,Y]]/12-B[X,B[X,B[X,B[X,Y]]]]/720+B[X,B[X,B[X,B[X,B[X,B[X,Y]]]]]]/30240-B[X,B[X,B[X,B[X,B[X,B[Y,B[X,Y]]]]]]]/20160-B[X,B[X,B[X,B[X,B[Y,B[X,Y]]]]]]/10080+B[X,B[X,B[X,B[X,B[Y,B[Y,B[X,Y]]]]]]]/20160-B[X,B[X,B[X,B[Y,B[X,Y]]]]]/1440+B[X,B[X,B[X,B[Y,B[X,B[X,B[X,Y]]]]]]]/12096-B[X,B[X,B[X,B[Y,B[X,B[Y,B[X,Y]]]]]]]/5040-B[X,B[X,B[X,B[Y,B[Y,B[X,Y]]]]]]/7560-(5*B[X,B[X,B[X,B[Y,B[Y,B[Y,B[X,Y]]]]]]])/24192-B[X,B[X,B[Y,B[X,Y]]]]/120+B[X,B[X,B[Y,B[X,B[X,B[X,Y]]]]]]/2520+B[X,B[X,B[Y,B[X,B[X,B[Y,B[X,Y]]]]]]]/3360+B[X,B[X,B[Y,B[X,B[Y,B[X,Y]]]]]]/3360+B[X,B[X,B[Y,B[X,B[Y,B[Y,B[X,Y]]]]]]]/2520-B[X,B[X,B[Y,B[Y,B[X,Y]]]]]/720-B[X,B[X,B[Y,B[Y,B[Y,B[X,Y]]]]]]/3360+B[X,B[X,B[Y,B[Y,B[Y,B[Y,B[X,Y]]]]]]]/20160-B[X,B[Y,B[X,Y]]]/24+B[X,B[Y,B[X,B[X,B[X,Y]]]]]/720-B[X,B[Y,B[X,B[X,B[X,B[X,B[X,Y]]]]]]]/20160-B[X,B[Y,B[X,B[X,B[X,B[Y,B[X,Y]]]]]]]/6720+B[X,B[Y,B[X,B[X,B[Y,B[X,Y]]]]]]/1680+B[X,B[Y,B[X,B[X,B[Y,B[Y,B[X,Y]]]]]]]/15120+B[X,B[Y,B[X,B[Y,B[X,Y]]]]]/240-B[X,B[Y,B[X,B[Y,B[X,B[X,B[X,Y]]]]]]]/10080-B[X,B[Y,B[X,B[Y,B[X,B[Y,B[X,Y]]]]]]]/2016+B[X,B[Y,B[X,B[Y,B[Y,B[X,Y]]]]]]/1260-B[X,B[Y,B[X,B[Y,B[Y,B[Y,B[X,Y]]]]]]]/20160-B[X,B[Y,B[Y,B[X,Y]]]]/360+B[X,B[Y,B[Y,B[X,B[X,B[Y,B[X,Y]]]]]]]/20160-B[X,B[Y,B[Y,B[X,B[Y,B[Y,B[X,Y]]]]]]]/10080+B[X,B[Y,B[Y,B[Y,B[X,Y]]]]]/1440+B[X,B[Y,B[Y,B[Y,B[Y,B[X,Y]]]]]]/10080-B[X,B[Y,B[Y,B[Y,B[Y,B[Y,B[X,Y]]]]]]]/60480-B[Y,B[X,Y]]/12+B[Y,B[X,B[X,B[X,Y]]]]/360-B[Y,B[X,B[X,B[X,B[X,B[X,Y]]]]]]/10080-B[Y,B[X,B[X,B[X,B[Y,B[X,Y]]]]]]/3360+B[Y,B[X,B[X,B[Y,B[Y,B[X,Y]]]]]]/7560+B[Y,B[X,B[Y,B[X,Y]]]]/120-B[Y,B[X,B[Y,B[X,B[X,B[X,Y]]]]]]/5040-B[Y,B[X,B[Y,B[X,B[Y,B[X,Y]]]]]]/1008-B[Y,B[X,B[Y,B[Y,B[Y,B[X,Y]]]]]]/10080+B[Y,B[Y,B[X,B[X,B[Y,B[X,Y]]]]]]/10080-B[Y,B[Y,B[X,B[Y,B[Y,B[X,Y]]]]]]/5040+B[Y,B[Y,B[Y,B[X,Y]]]]/720-B[Y,B[Y,B[Y,B[Y,B[Y,B[X,Y]]]]]]/30240;
Ev[X]=x; Ev[Y]=y;
Ev[B[u_, v_]] := Ev[B[u,v]] = b[Ev[u], Ev[v]];
Expand[x+y+(BCH8 /. B[u_,v_] :> Ev[B[u,v]])]
];

If[Head[$DegreeStack] =!= List,$DegreeStack={Infinity}];
$ToDegree=First[$DegreeStack];
SetAttributes[ToDegree,HoldRest];
ToDegree[m_,expr_] := Module[{res},
PrependTo[$DegreeStack,$ToDegree=m];
Clear[h]; h/: h^p_Integer /; p>$ToDegree := 0;
res=expr;
$DegreeStack=Rest[$DegreeStack];
$ToDegree=First[$DegreeStack];
Clear[h]; h/: h^p_Integer /; p>$ToDegree := 0;
res
];

PCollect[expr_] := Collect[expr, _ar, Expand];
b[ar[i_,j_], ar[k_,l_]] := Which[
i==k || i==j || k==l, 0,
j==l, h[i]ar[k,l] - h[k]ar[i,j],
j==k && i==l, -h[i]ar[j,i] + h[j]ar[i,j] - h[i]ar[j,j] + h[j]ar[i,i],
j==k, -h[i]ar[k,l] + h[k]ar[i,l],
i==l, -h[i]ar[k,j] + h[k]ar[i,j],
True, 0
];
b[0, y_] = 0; b[x_, 0] = 0;
b[x_Plus, y_] := PCollect[b[#,y]& /@ x];
b[x_, y_Plus] := PCollect[b[x,#]& /@ y];
b[c_*x_ar, y_] := PCollect[c*b[x,y]];
b[x_, c_*y_ar] := PCollect[c*b[x,y]];
Unprotect[NonCommutativeMultiply];
x_ ** y_ := BCH[b, x, y];

R[i_, j_] := h ar[i,j];

Unprotect[Up];
expr_ ~Up~ind_List := Module[
{ind1},
ind1 = Flatten[{#}]& /@ ind;
PCollect[expr /. {
h[i_] :> Total[h /@ ind1[[i]]],
ar[i_, j_] :> Total[Outer[ar, ind1[[i]], ind1[[j]]], 2]
}]
];

R4Eqn[V_] := V**(-R[1,3]-R[2,3])-(-R[1,3])**(-R[2,3])**V;
TwistEqn[V_] := (-V)**R[1,2]**(V~Up~{2,1})-(R[1,2]/2 + R[2,1]/2);
TwistMinusEqn[V_] := (-V)**(-R[2,1])**(V~Up~{2,1})+(R[1,2]/2 + R[2,1]/2);

Clear[Phi];
Phi[d_Integer, V_] := Phi[d, V] = PCollect[ToDegree[d,
(-V)~Up~{{1,2},3}**(-V)**V~Up~{2,3}**V~Up~{1,{2,3}}
]];
Phi[V_] := Phi[$ToDegree, V];

Pentagon[Phi_] := Phi**(Phi~Up~{1,{2,3},4})**(Phi~Up~{2,3,4})-(Phi~Up~{{1,2},3,4})**(Phi~Up~{1,2,{3,4}});
RR[i_,j_] := R[i,j]/2+R[j,i]/2;
Hexagon[Phi_] := RR[1,2]~Up~{{1,2},3} - Phi**RR[2,3]**((-Phi)~Up~{1,3,2})**RR[1,3]**(Phi~Up~{3,1,2});
HexagonMinus[Phi_] := (-RR[1,2])~Up~{{1,2},3} - Phi**(-RR[2,3])**((-Phi)~Up~{1,3,2})**(-RR[1,3])**(Phi~Up~{3,1,2});
uUnitarity[Phi_] := Phi**(Phi~Up~{3,2,1});

Puncture[inds___Integer, expr_] := expr /. Flatten[{h[#]->0, ar[#,_]->0}& /@ {inds}];

({{b[ar[1,2], ar[1,3]]}, {b[ar[1,3], ar[2,3]]}, {b[ar[1,2], ar[2,3]]}}) /. h -> c // MatrixForm