Pensieve header: One-Co computations in the abc presentation - testing notebook; continues pensieve://2015-08/.

SetDirectory["C:\\drorbn\\AcademicPensieve\\2015-11"];
<< abc.m

Testing Jacobi and Anti-Symmetry

VS=False;
AS[x1_,x2_] := Module[{as},
as=B[x1,x2]+B[x2,x1] // S;
If[as===0,as,{x1,x2} ->  as]
];
DeleteCases[Flatten[Outer[
AS,
FormalPlusBasis[3,f],
FormalPlusBasis[3,g]
]], 0]

AS[x1_,x2_] := Module[{as},
as=B[x1,x2]+B[x2,x1] // S;
If[as===0,as,{x1,x2} ->  as]
];
DeleteCases[Flatten[Outer[
AS,
FormalBasis[3,f],
FormalBasis[3,g]
]], 0]

JacPlusErrors=DeleteCases[
bas1=FormalPlusBasis[4,f];
bas2=FormalPlusBasis[4,g];
bas3=FormalPlusBasis[4,h];
Flatten[
Table[Jacobi[bas1[[i]],bas2[[j]],bas3[[k]]],
{i,Length[bas1]-1}, {j, i+1,Length@bas2}, {k,i+1,Length@bas3}]
],
0]

VS=False;
JacErrors=DeleteCases[
bas1=FormalBasis[4,f];
bas2=FormalBasis[4,g];
bas3=FormalBasis[4,h];
Flatten[
Table[Jacobi[bas1[[i]],bas2[[j]],bas3[[k]]],
{i,Length[bas1]-1}, {j, i+1,Length@bas2}, {k,i+1,Length@bas3}]
],
0]

Testing U2MM and MM2U

(#**𝟙)& /@ {a[f[(Subscript[b, 1]),(Subscript[b, 2]),(Subscript[b, 3])],1,2], a[f[(Subscript[b, 1]),(Subscript[b, 2]),(Subscript[b, 3])], 1,1]}

ff=f[(Subscript[b, 1]),(Subscript[b, 2]),(Subscript[b, 3]),(Subscript[b, 4])];
(#== MM2U[#**𝟙])& /@ {
a[ff,1,2], a[ff, 1,1], ca[ff, 1,2,3], ca[ff,2,2,3],δaa[ff,1,2,3,4], δaa[ff,1,1,2,3],δaa[ff,1,2,3,3], δaa[ff,1,1,2,2]
}

ff=f[(Subscript[b, 1]),(Subscript[b, 2]),(Subscript[b, 3]),(Subscript[b, 4])];
(#== MM2U[#]**𝟙)& /@ {
a1[ff,1,2], a1[ff, 1,1], ca1[ff,1,2,3],ca1[ff,2,2,3],δaa1[ff,1,2,3,4], δaa1[ff,1,1,2,3],δaa1[ff,1,2,3,3], δaa1[ff,1,1,2,2]
}

Testing (S^2)⊗(S^2)

VS1=False; ff=f[(Subscript[b, 1]),(Subscript[b, 2]),(Subscript[b, 3]),(Subscript[b, 4])];
DeleteCases[
Flatten[Table[
(δaa1[ff,i,j,k,l] // MM2U // S)==(δaa1[ff,i,j,k,l] // S // MM2U),
{i,4},{j,4},{k,4},{l,4}
]],
True
]

VS1=False; ff=f[(Subscript[b, 1]),(Subscript[b, 2]),(Subscript[b, 3]),(Subscript[b, 4])];
DeleteCases[
Flatten[Table[
δaa[ff,i,j,k,l]->((δaa[ff,i,j,k,l]**𝟙) // S)-((δaa[ff,i,j,k,l] // S)**𝟙),
{i,4},{j,4},{k,4},{l,4}
]],
_->0
]

Testing the Adjoint action

AutoAd[a[t,j,k]][a[1,k,j]]

AutoAd - Ad tests

Module[{t1,t2},
{t1=S[AutoAd[a[t,j,k]][#]],
S[Ad[a[t,j,k]][#] - t1]}
]& @ a[1,i,k]

AdTests[a[t,j,k]] = {β[f[(Subscript[b, j]),(Subscript[b, k])]], a[1,j,k],a[1,n,i],a[1,j,j],a[1,k,k],c[1, i], c[1, j],c[1, k],a[1,j,l],a[1,i,j],a[1,i,k], a[1,k,l], a[1,k,j]};

S[AutoAd[a[t,j,k]][#]-Ad[a[t,j,k]][#]]& /@ Take[AdTests[a[t,j,k]],All]

The semi group properties

Module[{t1,t2,t3,t4},
t1=Ad[a[t,j,k]][#] /. (h:(β|a|c|δa|ca|δaa))[c_,r___] :> h[SeriesCoefficient[c,{t,0,1}],r];
t2=B[a[1,j,k], #];
t3=# // Ad[a[t,j,k]] // Ad[a[s,j,k]];
t4=# // Ad[a[t+s,j,k]];
# -> S[{t1==t2,t3-t4}]
]& /@ AdTests[a[t,j,k]] // ColumnForm