Pensieve header: Testing å arrows.

SetDirectory["C:\\drorbn\\AcademicPensieve\\Projects\\OneCo-1604"];
<< Local-160529.m

Canonical forms for δåå’s

δåå[f_,i_,j_,k_,l_] := δaa[f,i,j,k,l]-δaa[(Subscript[b, i])f,ç,j,k,l]-δaa[(Subscript[b, k])f,i,j,ç,l];

Union[CF[UU[δaa[f,i,j,k,l]-δaa[f,i,l,k,j]] /. Thread[{i,j,k,l} -> #]]& /@ Tuples[{1,2},4]]

Union[CF[UU[δåå[f,i,j,k,l]-δåå[f,i,l,k,j]] /. Thread[{i,j,k,l} -> #]]& /@ Tuples[{1,2,3,4},4]]

Union[CF[UU[δåå[f,i,j,k,l]-δåå[f,Min@{i,k},Min@{j,l},Max@{i,k},Max@{j,l}]] /. Thread[{i,j,k,l} -> #]]& /@ Tuples[{1,2,3,4},4]]

Commuting heads within a δåå

{CF[UU[δaa[f,i,l,k,l]-δaa[f,k,l,i,l]]], CF[UU[δåå[f,i,l,k,l]-δåå[f,k,l,i,l]]]}

B=bb[1,2,3,4];

B[UU@a[1,1,2],UU@a[1,1,3]]

UU@δåå[1,1,2,1,3] // CF

UU@δåå[1,1,3,1,2] // CF

Calculations for OneCo-1605

Clear[ca,ac];
ac[f_,i_,j_,k_]:=ca[f,k,i,j]+(Subscript[Kδ, j,k])ao[f,i,j];
AutoCollecting /@ {c,ao,ca,aao};
ToOld=ReplaceAll[{c[f_,i_] :> δa[f,ç,i],
ao[f_,i_,j_]:> δa[f,i,j]-δa[(Subscript[b, i])f,ç,j],
ca[f_,i_,j_,k_]:>δaa[f,ç,i,j,k],
ac[f_,i_,j_,k_]:>δaa[f,i,j,ç,k],
aao[f_,i_,j_,k_,l_]:> δåå[f,i,j,k,l]
}];
ToNew=ReplaceAll[{
δa[f_,ç,j_]:>c[f,j],
δa[f_,i_,j_]:>ao[f,i,j]+c[(Subscript[b, i])f,j],
δaa[f_,ç,j_,k_,l_]:>ca[f,j,k,l],
δaa[f_,j_,k_,ç,l_]:>ac[f,j,k,l],
δaa[f_,i_,j_,k_,l_]:>aao[f,i,j,k,l]+ca[(Subscript[b, i])f,j,k,l]+ac[(Subscript[b, k])f,i,j,l]
}];

UU@aao[f,i,j,k,l]//ToOld // ToNew

DeleteCases[(#->CF[#-ToOld@ToNew@#])& /@ UUBasis[{1,2},{3,4},f],
_->UU[0]]

ac[f,i,j,k] = ca[f,k,i,j]+(Subscript[Kδ, j,k])ao[f,i,j]

UU@δaa[f,i,j,ç,k] // CF // ToNew

UU@δaa[f,i,j,ç,j] // CF // ToNew

NonCommutativeMultiply

t1=ToNew[UU[ao[f,i,j]//ToOld]**UU[a[g,k,l]//ToOld]]

UU[ca[f g (Subscript[b, k]),l,i,j]+aao[f g,i,j,k,l]]

t2=ToNew[UU[ao[f,i,j]//ToOld]**UU[a[g,j,l]//ToOld]]

t2-t1

tm, hm, hts, dm

δhb[f,z,ç,j]

UU@δhb[f,z,ç,j]//ToNew

UU@δhb[f,i,ç,y] // ToNew

UU[ca[f,y,j,x]] //ToOld// hm[x,y,z] // ToNew

UU[ca[f,x,j,y]] //ToOld// hm[x,y,z] // ToNew

UU[ca[f,x,j,x]] //ToOld// hm[x,y,z] // ToNew

UU[ca[f,y,j,y]] //ToOld// hm[x,y,z] // ToNew

UU[a[f[(Subscript[b, x]),(Subscript[b, y])],x,y]] // hts[y,x] // ToNew

UU[ao[f,x,y]] //ToOld// hts[y,x] // ToNew

UU@aao[f[(Subscript[b, x]),(Subscript[b, y])],i,j,k,l]//ToOld// hts[y,x] // ToNew

UU@aao[f[(Subscript[b, x]),(Subscript[b, y])],x,y,k,l]//ToOld// hts[y,x] // ToNew

UU@aao[f[(Subscript[b, x]),(Subscript[b, y])],x,j,k,y]//ToOld// hts[y,x] // ToNew

UU@aao[f[(Subscript[b, x]),(Subscript[b, y])],i,y,x,l]//ToOld// hts[y,x] // ToNew

UU@aao[f[(Subscript[b, x]),(Subscript[b, y])],i,j,x,y]//ToOld// hts[y,x] // ToNew

UU@aao[f[(Subscript[b, x]),(Subscript[b, y])],x,y,k,y]//ToOld// hts[y,x] // ToNew

UU@aao[f[(Subscript[b, x]),(Subscript[b, y])],i,y,x,y]//ToOld// hts[y,x] // ToNew

UU@aao[f[(Subscript[b, x]),(Subscript[b, y])],x,y,x,l]//ToOld// hts[y,x] // ToNew

UU@aao[f[(Subscript[b, x]),(Subscript[b, y])],x,j,x,y]//ToOld// hts[y,x] // ToNew

UU@aao[f[(Subscript[b, x]),(Subscript[b, y])],x,y,x,y]//ToOld// hts[y,x] // ToNew

tb, hb, thb, htb, db, bb on {β, a, δβ, ao, aao}

tb[x][ToOld@UU@a[f[(Subscript[b, x])],x,j],ToOld@UU@β[g[(Subscript[b, x])]]] // ToNew

tb[x][ToOld@UU@a[f[(Subscript[b, x])],x,j],ToOld@UU@a[g[(Subscript[b, x])],k,l]] // ToNew

tb[x][ToOld@UU@a[f[(Subscript[b, x])],x,j],ToOld@UU@a[g[(Subscript[b, x])],x,l]] // ToNew

tb[x][ToOld@UU@a[f[(Subscript[b, x])],j,k],ToOld@UU@a[g,x,l]] // ToNew

tb[x][ToOld@UU@a[f[(Subscript[b, x])],x,k],ToOld@UU@a[g,x,l]] // ToNew

hb[y][ToOld@UU@a[f,i,y],ToOld@UU@c[g,y]] // ToNew

hb[y][ToOld@UU@a[f,i,y],ToOld@UU@ao[g,j,y]] // ToNew

hb[y][ToOld@UU@a[f,i,y],ToOld@UU@ca[g,j,k,l]] // ToNew

hb[y][ToOld@UU@a[f,i,y],ToOld@UU@ca[g,y,k,l]] // ToNew

hb[y][ToOld@UU@a[f,i,y],ToOld@UU@ca[g,j,k,y]] // ToNew

hb[y][ToOld@UU@a[f,i,y],ToOld@UU@ca[g,y,k,y]] // ToNew

hb[y][ToOld@UU@a[f,i,y],ToOld@UU@aao[g,j,y,l,m]] // ToNew

hb[y][ToOld@UU@a[f,i,y],ToOld@UU@aao[g,j,k,l,y]] // ToNew

hb[y][ToOld@UU@a[f,i,y],ToOld@UU@aao[g,j,y,l,y]] // ToNew

thb

thb[x,y][ToOld@UU@a[f[(Subscript[b, x])],i,j], ToOld@UU@a[g,k,l]] // ToNew

t1=thb[x,y][ToOld@UU@a[f[(Subscript[b, x])],i,j], ToOld@UU@a[g,k,y]] // ToNew

t2=thb[x,y][ToOld@UU@a[f[(Subscript[b, x])],x,j], ToOld@UU@a[g,k,y]] // ToNew

t2-t1

CF[t2-t1 /. i->x]

thb[x,y][ToOld@UU@a[f,x,j], ToOld@UU@c[g,y]] // ToNew

thb[x,y][ToOld@UU@a[f,x,j], ToOld@UU@ao[g,k,y]] // ToNew

thb[x,y][ToOld@UU@a[f,x,j], ToOld@UU@ca[g,k,l,m]] // ToNew

t1=thb[x,y][ToOld@UU@a[f,x,j], ToOld@UU@ca[g,y,l,m]] // ToNew

t2=thb[x,y][ToOld@UU@a[f,x,j], ToOld@UU@ca[g,k,l,y]] // ToNew

t3=thb[x,y][ToOld@UU@a[f,x,j], ToOld@UU@ca[g,y,l,y]] // ToNew

CF[(t3-t1-t2) /. {k|m->y}]

thb[x,y][ToOld@UU@a[f,x,j], ToOld@UU@aao[g,k,l,m,n]] // ToNew

t1=thb[x,y][ToOld@UU@a[f,x,j], ToOld@UU@aao[g,k,y,m,n]] // ToNew

t2=thb[x,y][ToOld@UU@a[f,x,j], ToOld@UU@aao[g,k,l,m,y]] // ToNew

t3=thb[x,y][ToOld@UU@a[f,x,j], ToOld@UU@aao[g,k,y,m,y]] // ToNew

CF[(t3-t1-t2) /. {l|n->y}]

thb[x,y][ToOld@UU@ao[f,x,j], ToOld@UU@a[g,k,y]] // ToNew

thb[x,y][ToOld@UU@ca[f,m,x,j], ToOld@UU@a[g,k,y]] // ToNew

thb[x,y][ToOld@UU@aao[f,i,j,m,n], ToOld@UU@a[g,k,y]] // ToNew

t1=thb[x,y][ToOld@UU@aao[f,x,j,m,n], ToOld@UU@a[g,k,y]] // ToNew

t2=thb[x,y][ToOld@UU@aao[f,i,j,x,n], ToOld@UU@a[g,k,y]] // ToNew

t3=thb[x,y][ToOld@UU@aao[f,x,j,x,n], ToOld@UU@a[g,k,y]] // ToNew

CF[t3-t1-t2 /. i|m->x]

ct (contract)

ct[h,t][ToOld@UU@a[f,i,h], ToOld@UU@ao[g,t,j]] //ToNew

ct[h,t][ToOld@UU@a[f,i,h], ToOld@UU@ao[g[(Subscript[b, t])],j,k]] //ToNew

ct[h,t][ToOld@UU@ao[f,i,h], ToOld@UU@β[g[(Subscript[b, t])]]] //ToNew

pp[a[f,i,h], ao[g,j,k]]//ToNew[ct[h,t][ToOld@UU@#[[1]], ToOld@UU@(#[[2]]/.g->g[(Subscript[b, t])])]]&

pp[ao[f,i,h], a[g,j,k]]//ToNew[ct[h,t][ToOld@UU@#[[1]], ToOld@UU@(#[[2]]/.g->g[(Subscript[b, t])])]]&

pp[aao[f,i,h,j,k], β[g]]//ToNew[ct[h,t][ToOld@UU@#[[1]], ToOld@UU@(#[[2]]/.g->g[(Subscript[b, t])])]]&

pp[aao[f,i,h,j,k], a[g,l,m]]//ToNew[ct[h,t][ToOld@UU@#[[1]], ToOld@UU@(#[[2]]/.g->g[(Subscript[b, t])])]]&

pp[aao[f,i,j,k,h], β[g]]//ToNew[ct[h,t][ToOld@UU@#[[1]], ToOld@UU@(#[[2]]/.g->g[(Subscript[b, t])])]]&

pp[aao[f,i,j,k,h], a[g,l,m]]//ToNew[ct[h,t][ToOld@UU@#[[1]], ToOld@UU@(#[[2]]/.g->g[(Subscript[b, t])])]]&

pp[c[f,h], a[g,j,k]]//ToNew[ct[h,t][ToOld@UU@#[[1]], ToOld@UU@(#[[2]]/.g->g[(Subscript[b, t])])]]&

pp[ca[f,h,i,j], β[g]]//ToNew[ct[h,t][ToOld@UU@#[[1]], ToOld@UU@(#[[2]]/.g->g[(Subscript[b, t])])]]&

pp[ca[f,h,i,j], a[g,k,l]]//ToNew[ct[h,t][ToOld@UU@#[[1]], ToOld@UU@(#[[2]]/.g->g[(Subscript[b, t])])]]&