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Generate.m
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Generate.m
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BeginPackage["SuperLie`Generate`", {"SuperLie`", "SuperLie`Domain`"}]
SuperLie`Generate`GenBasis::usage =
"GenBasis[g] is the representation of the basis of \"g\" in terms of
(free) generators.";
SuperLie`Generate`GenRel::usage =
"GenRel[g] is the list of relations between the generators of \"g\"."
SuperLie`Generate`ToDegree::usage =
"ToDegree->n is an optional parameter for space constructors. It restricts the
calculations to the elements of polynomial degree d in terms of generators.
ToDegree[m] returns this limit. The results of operations in m are evaluated
only if its degree <= ToDegree[m]."
(*
ToDegree::usage =
"Option ToDegree->d restrict the calculation in algebra constructors to the
elements of polynomial degree d in terms of generators."
*)
SuperLie`Generate`GRange::usage = "Replaced by ToDegree."
SuperLie`Generate`GRange = ToDegree;
(*SuperLie`Generate`JacRel::usage =
"JacRel[g] is the list of Jacobi relations between the generators of \"g\"."*)
Begin["$`"]
NewValue[GenBasis, GenRel, ToDegree]
(* ----------- Step of generation ----------------------------------------- *)
(* 960630: new posibitity: restriction for Grade[g[i]] *)
StepGeneration[g_, deg_, rng_, Brk_, brk_, sqr_:False] :=
Module[{ q, s, t, vars, jac, sol, solj={}, prevtmp, hl, i, j, k, l, mgr},
(* extend the table of commutators *)
Clear[gen$var];
VectorQ[gen$var]^=True;
If [deg==1, Return[0]];
With[{imin=gen$ind[[deg-1]], imax=gen$ind[[deg]]-1,
jmax=If [2*(deg-1)<=rng, i, gen$ind[[rng-deg+2]]-1]},
gen$tbl = Join[gen$tbl, Table[None, {i, imin, imax }, {j, jmax} ]];
If[sqr, gen$sqr = Join[gen$sqr, Table[None, {i, imin, imax }]]]
];
(* the list "gt" of pairs [g[j],g[i]] of degree "r" *)
gen$lst = prevtmp = gen$flag = {};
For [q=1; l=0, q<=deg/2, ++q,
s = deg-q;
For [i=gen$ind[[q]], i<gen$ind[[q+1]], ++i,
If[NumberQ[maxGrade$], mgr = maxGrade$-Grade[g[i]]];
For [j=If[q==s, i, gen$ind[[s]]], j<gen$ind[[s+1]], ++j,
If [i==j && P[g[i]]==0, gen$tbl[[i,i]] = 0; Continue[] ];
If [NumberQ[maxGrade$] && (Grade[g[j]]>mgr), Continue[] ];
prevtmp = {prevtmp, i}; (* precedence for Holl basis *)
If [sqr && i==j,
gen$lst = { gen$lst, Squaring[g[i],brk] }; (* name list *)
hl = 1;
gen$sqr[[i]] = gen$var[hl, ++l];
gen$tbl[[j,i]] = SVTimes[2, gen$var[hl, l]],
(*else*)
gen$lst = { gen$lst, brk[g[i],g[j]] }; (* name list *)
hl = If[i>=gen$prev[[j]], 2, 0];
gen$tbl[[j,i]] = gen$var[hl, ++l]
];
gen$flag = {gen$flag, hl};
] ] ];
DPrint[4, "Added ", l, " pair(s)"];
DPrint[5, "Added ", Flatten[gen$lst]];
If [l==0, Return[0]]; (* No more elements *)
gen$lst = Flatten[gen$lst];
gen$flag = Flatten[gen$flag];
prevtmp = Flatten[prevtmp];
(* Equations from Jacobi identities *)
For [q=1, q<=deg/3, ++q,
For[s=q, s<=(deg-q)/2, ++s,
t = deg-q-s;
Do [jac = VNormal[ VPlus[ Brk[g[i],Brk[g[j],g[k]]],
(-1) ~SVTimes~ Brk[ Brk[g[i],g[j]], g[k] ],
(-(-1)^(P[g[i]]*P[g[j]]))~SVTimes~Brk[g[j],Brk[g[i],g[k]]] ] /.
solj ];
vars = MatchList[jac, _gen$var];
If [vars=={}, Continue[] ];
DPrint[5, "Solving ", jac==0, " # ", {i,j,k}->{g[i],g[j],g[k]}];
sol = VSolve[jac==0, {vars[[1]]}, InverseFunctions->True, VerifySolutions->False] [[1]];
(* sol = VSolve[jac==0, {vars[[1]]}] [[1]]; *)
DPrint[5, "Solution ", sol ];
If [sol=!={}, gen$flag[[ sol[[1,1,-1]] ]] = True;
sol = MapAt[VNormal, sol[[1]], 2 ];
solj = Append[ MapAt[VNormal,#/.sol,2]& /@ solj, sol];
],
{i, gen$ind[[q]], gen$ind[[q+1]]-1},
{j, Max[i,gen$ind[[s]]], gen$ind[[s+1]]-1},
{k, Max[j,gen$ind[[t]]], gen$ind[[t+1]]-1}
]
] ];
(* Extra Jacobi identies for char=2 *)
If[sqr && $p==2,
For[s=1, s<=(deg-1)/2, ++s,
q = deg-2*s;
Do [ If[P[g[j]]==1,
jac = VNormal[
VPlus[
Brk[g[i],Squaring[g[j],Brk]],
(-1) ~SVTimes~ Brk[g[j], Brk[g[j], g[i]]]
] /. solj ];
vars = MatchList[jac, _gen$var];
If [vars=={}, Continue[] ];
DPrint[5, "Solving (sqr)", jac==0, {i,j}->{g[i],g[j]} ];
sol = VSolve[jac==0, {vars[[1]]}, InverseFunctions->True, VerifySolutions->False] [[1]];
(* sol = VSolve[jac==0, {vars[[1]]}] [[1]]; *)
DPrint[5, "Solution ", sol ];
If [sol=!={}, gen$flag[[ sol[[1,1,-1]] ]] = True;
sol = MapAt[VNormal, sol[[1]], 2 ];
solj = Append[ MapAt[VNormal,#/.sol,2]& /@ solj, sol];
]],
{i, gen$ind[[q]], gen$ind[[q+1]]-1},
{j, Max[i,gen$ind[[s]]], gen$ind[[s+1]]-1}
]
] ];
(* Extra Jacobi identies for char=3 *)
If[$p==3 && Mod[deg,3]==0,
q = deg/3;
Do [ If[P[g[i]]==1,
jac = VNormal[Brk[g[i],If[sqr, Squaring[g[i],Brk], Brk[g[i],g[i]]]] /. solj ];
vars = MatchList[jac, _gen$var];
If [vars=={}, Continue[] ];
DPrint[5, "Solving (cub)", jac==0, {i}->{g[i]} ];
sol = VSolve[jac==0, {vars[[1]]}, InverseFunctions->True, VerifySolutions->False] [[1]];
(* sol = VSolve[jac==0, {vars[[1]]}] [[1]]; *)
DPrint[5, "Solution ", sol ];
If [sol=!={}, gen$flag[[ sol[[1,1,-1]] ]] = True;
sol = MapAt[VNormal, sol[[1]], 2 ];
solj = Append[ MapAt[VNormal,#/.sol,2]& /@ solj, sol];
]],
{i, gen$ind[[q]], gen$ind[[q+1]]-1}
]
];
(* gen$tbl = gen$tbl //. solj; *)
Apply[Set, solj, {1}];
gen$tbl = gen$tbl;
If[sqr, gen$sqr = gen$sqr];
DPrint[1, "npairs = ",l," (",N[TimeUsed[ ]-$tm,4], " sec)"]; $tm = TimeUsed[];
Return[l]
]
(* version with generators with different gradings *)
StepGenerationVG[g_, grade_, deg_, rng_, Brk_, brk_, sqr_:False] :=
Module[{ q, s, t, vars, jac, sol, solj={}, prevtmp, hl, i, j, k, l, mgr, g1, g2, g3},
(* extend the tables of commutators and squares*)
Clear[gen$var];
VectorQ[gen$var]^=True;
If [deg==1, Return[0]];
(* the list "gt" of pairs [g[j],g[i]] of degree "deg" *)
gen$lst = prevtmp = gen$flag = {};
For[g1=gr$min; l=0, g1<=Min[grade/2,grade-gr$min], g1++,
g2 = grade - g1;
(*If[grade>=2, DPrint[1,"Trying grades g1=",g1,", g2=",g2]];*)
For [q=Max[1,deg-gen$rng[g2]], q<=Min[gen$rng[g1], If[g1==g2, deg/2, deg-1]], ++q,
s = deg-q;
(*If[grade>=2, DPrint[1,"Trying degrees d1=",q,", d2=",s]];*)
For [i=gen$ind[g1][[q]], i<gen$ind[g1][[q+1]], ++i,
For [j=If[g1==g2 && q==s, i, gen$ind[g2][[s]]], j<gen$ind[g2][[s+1]], ++j,
If [i==j && P[g[i]]==0, gen$tbl[[i,i]] = 0; Continue[] ];
prevtmp = {prevtmp, i}; (* precedence for Holl basis *)
If[j>Length[gen$tbl], (* extend the tables if required *)
gen$tbl = Join[gen$tbl, Table[None, {ii, Length[gen$tbl]+1, j+25}, {ii}]];
If[sqr, gen$sqr = Join[gen$sqr, Table[None, {ii, Length[gen$sqr]+1, j+25}]]]];
If [sqr && i==j,
gen$lst = { gen$lst, Squaring[g[i],brk] }; (* name list *)
hl = 1;
gen$sqr[[i]] = gen$var[hl, ++l];
gen$tbl[[j,i]] = SVTimes[2, gen$var[hl, l]],
(*else*)
gen$lst = { gen$lst, brk[g[i],g[j]] }; (* name list *)
hl = If[i>=gen$prev[[j]], 2, 0];
gen$tbl[[j,i]] = gen$var[hl, ++l]];
gen$flag = {gen$flag, hl};
] ] ] ];
DPrint[4, "Added ", l, " pair(s)"];
If [l==0, Return[0]]; (* No more elements *)
gen$lst = Flatten[gen$lst];
gen$flag = Flatten[gen$flag];
prevtmp = Flatten[prevtmp];
(* Equations from Jacobi identities *)
For[g1=gr$min, g1<=Min[grade/3,grade-2*gr$min], g1++,
For[g2=g1, g2<=Min[(grade-g1)/2,grade-g1-gr$min], g2++,
g3 = grade-g1-g2;
For [q=1, q<=Min[gen$rng[g1],Which[g1==g3, deg/3, g1==g2, (deg-1)/2, True, deg-2]], ++q,
For[s=q, s<=Min[gen$rng[g2],If[g2==g3, (deg-q)/2, deg-q-1]], ++s,
t = deg-q-s;
If[t<=gen$rng[g3],
Do [jac = VNormal[ VPlus[ Brk[g[i],Brk[g[j],g[k]]],
(-1) ~SVTimes~ Brk[ Brk[g[i],g[j]], g[k] ],
(-(-1)^(P[g[i]]*P[g[j]]))~SVTimes~Brk[g[j],Brk[g[i],g[k]]] ] /.
solj ];
vars = MatchList[jac, _gen$var];
If [vars=={}, Continue[] ];
DPrint[5, "Solving ", jac==0 ];
sol = VSolve[jac==0, {vars[[1]]}, InverseFunctions->True, VerifySolutions->False] [[1]];
DPrint[5, "Solution ", sol ];
If [sol=!={}, gen$flag[[ sol[[1,1,-1]] ]] = True;
sol = MapAt[VNormal, sol[[1]], 2 ];
solj = Append[ MapAt[VNormal,#/.sol,2]& /@ solj, sol];
],
{i, gen$ind[g1][[q]], gen$ind[g1][[q+1]]-1},
{j, Max[i,gen$ind[g2][[s]]], gen$ind[g2][[s+1]]-1},
{k, Max[j,gen$ind[g3][[t]]], gen$ind[g3][[t+1]]-1}
]]
]]]];
(* Extra Jacobi identies for char=2 *)
If[sqr && $p==2,
For[g2=gr$min, g2<=(grade-gr$min)/2, g2++,
g1 = grade-2*g2;
For [s=1, s<=Min[gen$rng[g2],(deg-1)/2], ++s,
For[s=q, s<=Min[gen$rng[g2],If[g2==g3, (deg-q)/2, deg-q-1]], ++s,
q = deg-2*s;
If[q<=gen$rng[g1],
Do [ If[P[g[j]]==1,
jac = VNormal[
VPlus[
Brk[g[i],Squaring[g[j],Brk]],
(-1) ~SVTimes~ Brk[g[j], Brk[g[j], g[i]]]
] /. solj ];
vars = MatchList[jac, _gen$var];
If [vars=={}, Continue[] ];
DPrint[5, "Solving ", jac==0 ];
sol = VSolve[jac==0, {vars[[1]]}, InverseFunctions->True, VerifySolutions->False] [[1]];
DPrint[5, "Solution ", sol ];
If [sol=!={}, gen$flag[[ sol[[1,1,-1]] ]] = True;
sol = MapAt[VNormal, sol[[1]], 2 ];
solj = Append[ MapAt[VNormal,#/.sol,2]& /@ solj, sol];
]],
{i, gen$ind[g1][[q]], gen$ind[g1][[q+1]]-1},
{j, Max[i,gen$ind[g2][[s]]], gen$ind[g2][[s+1]]-1}
]]
]]]];
(* Extra Jacobi identies for char=3 *)
If[$p==3 && Mod[deg,3]==0 && Mod[grade,3]==0,
g1=grade/3;
q = deg/3;
Do [ If[P[g[i]]==1,
jac = VNormal[Brk[g[i],If[sqr, Squaring[g[i],Brk], Brk[g[i],g[i]]]] /. solj ];
vars = MatchList[jac, _gen$var];
If [vars=={}, Continue[] ];
DPrint[5, "Solving (cub)", jac==0, {i}->{g[i]} ];
sol = VSolve[jac==0, {vars[[1]]}, InverseFunctions->True, VerifySolutions->False] [[1]];
(* sol = VSolve[jac==0, {vars[[1]]}] [[1]]; *)
DPrint[5, "Solution ", sol ];
If [sol=!={}, gen$flag[[ sol[[1,1,-1]] ]] = True;
sol = MapAt[VNormal, sol[[1]], 2 ];
solj = Append[ MapAt[VNormal,#/.sol,2]& /@ solj, sol];
]],
{i, gen$ind[g1][[q]], gen$ind[g1][[q+1]]-1}
]
];
(* gen$tbl = gen$tbl //. solj; *)
Apply[Set, solj, {1}];
gen$tbl = gen$tbl;
If[sqr, gen$sqr = gen$sqr];
DPrint[1, "npairs = ",l," (",N[TimeUsed[ ]-$tm,4], " sec)"]; $tm = TimeUsed[];
Return[l]
]
(* --------- Extension of the the basis --------------- *)
StepExtendBasis[g_, npairs_,sqr_:False] :=
Module [{ i, rdim, dim, rng, par={} },
rng = Length[gen$ind];
dim = gen$ind[[-1]]-1;
For [ i=1, i<=npairs, ++i, (* add gen$var[i] to basis *)
If [ gen$flag[[i]], Continue[] ]; (* gen$var[i] was removed *)
gen$basis = {gen$basis, gen$lst[[i]]};
If [gen$super, par = {par, P[ gen$lst[[i]] ]} ];
gen$prev = {gen$prev, gen$lst[[i,1,1]]};
gen$var[_,i] = g[++dim]
];
gen$tbl = gen$tbl//.RestoreSV; (* replace all gen$var[i] *)
If[sqr, gen$sqr = gen$sqr//.RestoreSV];
gen$rel = gen$rel//.RestoreSV;
Clear[gen$var];
VectorQ[gen$var]^=True;
AppendTo[gen$ind, dim+1];
rdim = gen$ind[[-1]] - gen$ind[[-2]];
If [rdim>0,
gen$rng = rng;
gen$deg = Join[gen$deg, Table[rng, {rdim}]];
gen$prev = Flatten[gen$prev];
];
If [gen$super,
gen$par = Join[gen$par, Flatten[par]];
i = Plus @@ gen$par;
DPrint[1, "r = ", rng, ", Dim = (", dim-i, "|", i, ")" ],
(*else*)
DPrint[1, "r = ", rng, ", Dim = ", dim]
];
dim
]
End[]
EndPackage[]