Pensieve header: Testing notebook for Scatter and Glow in OneCo. Continues pensieve://2016-02/.

In the U(T)U(H) conventions.

SetDirectory["C:\\drorbn\\AcademicPensieve\\2016-03"];
<< SnG.m

Bases

UUBasis[{1,2},{3,4},f]

Meta-associativity for tm

((# // tm[1,2,1] // tm[1,3,1]) == (# // tm[2,3,2] // tm[1,2,1]))& /@ UUBasis[{1,2,3,4},{5,6},f] // Union

Meta-associativity for hm

((# // hm[1,2,1] // hm[1,3,1]) == (# // hm[2,3,2] // hm[1,2,1]))& /@ UUBasis[{5,6},{1,2,3,4},f] // Union

Compatibility between tm and hts

tmhts[u_] := (u// tm[1,2,1] // hts[4,1]) -(u// hts[4,1] // hts[4,2] // tm[1,2,1]);
DeleteCases[
{#} -> tmhts[#]& /@ UUBasis[{1,2,3},{4,5},f],
_ -> UU[0]
]

tmhts[u_] := (u// tm[1,2,1] // hts[5,1]) -(u// hts[5,1] // hts[5,2] // tm[1,2,1]);
DeleteCases[
{#} -> tmhts[#]& /@ UUBasis[{1,2,3},{4,5},f],
_ -> UU[0]
]

tmhts[u_] := (u// tm[2,1,1] // hts[4,1]) -(u// hts[4,2] // hts[4,1] // tm[2,1,1]);
DeleteCases[
{#} -> tmhts[#]& /@ UUBasis[{1,2,3},{4,5},f],
_ -> UU[0]
]

tmhts[u_] := (u// tm[2,1,1] // hts[5,1]) -(u// hts[5,2] // hts[5,1] // tm[2,1,1]);
DeleteCases[
{#} -> tmhts[#]& /@ UUBasis[{1,2,3},{4,5},f],
_ -> UU[0]
]

Compatibility between hm and hts

hmhts[u_] := (u// hm[1,2,1] // hts[1,4]) -(u// hts[2,4] // hts[1,4] // hm[1,2,1]);
DeleteCases[
{#} -> hmhts[#]& /@ UUBasis[{4,5},{1,2,3},f],
_ -> UU[0]
]

hmhts[u_] := (u// hm[1,2,1] // hts[1,5]) -(u// hts[2,5] // hts[1,5] // hm[1,2,1]);
DeleteCases[
{#} -> hmhts[#]& /@ UUBasis[{4,5},{1,2,3},f],
_ -> UU[0]
]

hmhts[u_] := (u// hm[2,1,1] // hts[1,4]) -(u// hts[1,4] // hts[2,4] // hm[2,1,1]);
DeleteCases[
{#} -> hmhts[#]& /@ UUBasis[{4,5},{1,2,3},f],
_ -> UU[0]
]

hmhts[u_] := (u// hm[2,1,1] // hts[1,5]) -(u// hts[1,5] // hts[2,5] // hm[2,1,1]);
DeleteCases[
{#} -> hmhts[#]& /@ UUBasis[{4,5},{1,2,3},f],
_ -> UU[0]
]

Meta-associativity for dm

((# // dm[1,2,1] // dm[1,3,1]) == (# // dm[2,3,2] // dm[1,2,1]))& /@ UUBasis[4,f] // Union

Meta-AS and meta-Jacobi for tb

tbAS[u_,v_] := tb[0][u, v] + tb[0][v,u];
DeleteCases[
Flatten[Outer[
{##} -> tbAS[##]&,
UUBasis[{0,1},{1,2}, f], UUBasis[{0,2},{3,4},g]
]],
_ -> UU[0]
]

tbJacobi[u_,v_,w_] := tb[0][u, tb[0][v,w]] + tb[0][v, tb[0][w,u]] + tb[0][w, tb[0][u,v]];
DeleteCases[
Flatten[Outer[
{##} -> tbJacobi[##]&,
UUBasis[{0,1},{1,2}, f], UUBasis[{0,2},{3,4},g], UUBasis[{0,3},{5,6},h]
]],
_ -> UU[0]
]

Compatibility of tb and tm

tbtm[u_,v_] := Plus[
tb[1][u,tσ[2,-2][v]] // tm[2,-2,2],
-tb[1][u,tσ[2,-2][v]] // tm[-2,2,2],
-tb[2][u,tσ[1,-1][v]] // tm[1,-1,1],
tb[2][u,tσ[1,-1][v]] // tm[-1,1,1]
];
DeleteCases[
Flatten[Outer[
{##} -> tbtm[##]&,
UUBasis[{1,2,3},{1,2}, f], UUBasis[{1,2,4},{3,4},f]
]],
_ -> UU[0]
]

Meta-AS and meta-Jacobi for hb

hbAS[u_,v_] := hb[0][u, v] + hb[0][v,u];
DeleteCases[
Flatten[Outer[
{##} -> hbAS[##]&,
UUBasis[{1,2},{0,1}, f], UUBasis[{3,4},{0,2},g]
]],
_ -> UU[0]
]

hbJacobi[u_,v_,w_] := hb[0][u, hb[0][v,w]] + hb[0][v, hb[0][w,u]] + hb[0][w, hb[0][u,v]];
DeleteCases[
Flatten[Outer[
{##} -> hbJacobi[##]&,
UUBasis[{1,2},{0,1}, f], UUBasis[{3,4},{0,2},g], UUBasis[{5,6},{0,3},h]
]],
_ -> UU[0]
]

Compatibility of hb and hm

hbhm[u_,v_] := Plus[
hb[1][u,hσ[2,-2][v]] // hm[2,-2,2],
-hb[1][u,hσ[2,-2][v]] // hm[-2,2,2],
-hb[2][u,hσ[1,-1][v]] // hm[1,-1,1],
hb[2][u,hσ[1,-1][v]] // hm[-1,1,1]
];
DeleteCases[
Flatten[Outer[
{##} -> hbhm[##]&,
UUBasis[{1,2},{1,2,3}, f], UUBasis[{3,4},{1,2,4},g]
]],
_ -> UU[0]
]

Meta-Jacobi for thb/tb and for thb/hb

tthJacobi[u_,v_,w_] := Plus[
-thb[0,0][tb[0][u,v],w]+tb[0][thb[0,0][u,w],v]-thb[0,0][v, thb[0,0][u,w]]+tb[0][u,thb[0,0][v,w]]+thb[0,0][u,thb[0,0][v,w]]
];
DeleteCases[
Flatten[Outer[
{##} -> tthJacobi[##]&,
UUBasis[{0,1},{1,2}, f], UUBasis[{0,2},{3,4},g], UUBasis[{3,4},{0,5},h]
]],
_ -> UU[0]
]

thhJacobi[u_,v_,w_] := Plus[
-thb[0,0][u,hb[0][v,w]]+hb[0][thb[0,0][u,v],w]+thb[0,0][thb[0,0][u,v], w]+hb[0][v,thb[0,0][u,w]]-thb[0,0][thb[0,0][u,w],v]
];
DeleteCases[
Flatten[Outer[
{##} -> thhJacobi[##]&,
UUBasis[{0,1},{1,2}, f], UUBasis[{2,3},{0,3},g], UUBasis[{4,5},{0,4},h]
]],
_ -> UU[0]
]

Meta-AS and meta-Jacobi for db

dbAS[u_,v_] := db[0][u, v] + db[0][v,u];
DeleteCases[
Flatten[Outer[
{##} -> dbAS[##]&,
UUBasis[{0,1,2}, f], UUBasis[{0,3,4},g]
]],
_ -> UU[0]
]

dbJacobi[u_,v_,w_] := db[0][u, db[0][v,w]] + db[0][v, db[0][w,u]] + db[0][w, db[0][u,v]];
Module[{len, bas1, bas2, bas3,test},
len=Length[bas1=UUBasis[{0,1,2}, f]];
bas2=UUBasis[{0,3,4},g];
bas3=UUBasis[{0,5,6},h];
DeleteCases[
Flatten[Table[
test={bas1[[i]],bas2[[j]],bas3[[k]]};
test -> dbJacobi@@test,
{i,1,len-2}, {j,i+1,len-1},{k,j+1,len}
]],
_ -> UU[0]
]
]

Compatibility of db and dm

dbdm[u_,v_] := Plus[
db[1][u,dσ[2,-2][v]] // dm[2,-2,2],
-db[1][u,dσ[2,-2][v]] // dm[-2,2,2],
-db[2][u,dσ[1,-1][v]] // dm[1,-1,1],
db[2][u,dσ[1,-1][v]] // dm[-1,1,1]
];
DeleteCases[
Flatten[Outer[
{##} -> dbdm[##]&,
UUBasis[{1,2,3}, f], UUBasis[{1,2,4},g]
]],
_ -> UU[0]
]

AS and Jacobi for bb

bb4=bb[1,2,3,4];
bbAS[u_,v_] := bb4[u, v] + bb4[v,u];
DeleteCases[
Flatten[Outer[
{##} -> bbAS[##]&,
UUBasis[2, f], UUBasis[2,g]
]],
_ -> UU[0]
]

bb3=bb[1,2,3];
bbJacobi[u_,v_,w_] := bb3[u, bb3[v,w]] + bb3[v, bb3[w,u]] + bb3[w, bb3[u,v]];
Module[{bas1,bas2,bas3,len,test},
{bas1,bas2,bas3} ={UUBasis[3, f], UUBasis[3,g], UUBasis[3,h]};
len=Length[bas1];
DeleteCases[
Flatten[Table[
test={bas1[[i]],bas2[[j]],bas3[[k]]};
test -> bbJacobi@@test,
{i,1,len-2}, {j,i+1,len-1},{k,j+1,len}
]],
_ -> UU[0]
]
]

ct (contract) poperties

(Outer[
ct[#1 // tm[1,2,1], #2]-(ct[#1, #2] // tm[1,2,1])&,
UUBasis[{1,2,3},{0,1},f], UUBasis[{0,4},{2,3},g]
] //Flatten// Union) /. (Subscript[ϵ, 4])->1

Outer[
{#1,#2} -> Equal[
ct[#1 // hm[1,2,1], #2],
ct[#1, #2] // hm[1,2,1]
]&,
UUBasis[{1,2},{0,1,2,3},f], UUBasis[{0,4},{4,5},g]
] //Flatten // DeleteCases[#, _->True]&

Outer[
{#1,#2} -> Equal[
ct[#1, #2 // hm[2,3,2]],
ct[#1, #2] // hm[2,3,2]
]&,
UUBasis[{1,2},{0,1},f], UUBasis[{0,3},{2,3,4},g]
] //Flatten // DeleteCases[#, _->True]&

Outer[
{#1,#2} -> Equal[
ct[#1, #2 // tm[3,4,3]],
ct[#1, #2] // tm[3,4,3]
]&,
UUBasis[{1,2},{0,1},f], UUBasis[{0,3,4,5},{2,3},g]
] //Flatten // DeleteCases[#, _->True]&