LeibnizQuasiCover.gi

##################################################
InstallGlobalFunction(LeibnizQuasiCoveringHomomorphism,
function(L)
local 
      BasisL, lenBL, SCTL, lisabelian, K, SCTC, u1, u2, u3, u4, l1, l2,
      index1, index2, C, t1, t2, t3, t4, t5, t6, i, j, k, m, p,
      vectorsI, vectorsII,w1,w2,
      zr1, mr1, zr2, mr2, derayeh1, derayeh2, derayeh3, derayeh4,
      derayeh5, derayeh6, derayeh7, derayeh8, derayeh9, derayeh10, derayeh11, derayeh12,
      lenv1, lenv2, lenv3, lenv4, lenv5, lenv6, 
      e, q, w, v1, v2, lzr1, lzr2, lmr1, lmr2,
      q1, q2, tt1, tt2, ww, h, LTL, LVL, t, v7, I, II,
      bL, bLTL, g, MLTL, v, vv1, BI, I1, I2, I3,
      LenBI, BLTL, BBLTL, liltl, llzr1,
      II1,II2,II3,BII,LenBII,
      LL,f,L2,dlab,nn,Tens,pif,BL2,l,vpif,Bvpif,lenbvpif,sp,KK,
      imagesp,bimagesp,MLAB,LenBL2,BLAB,BBLAB,IIII,LAB,BL,pi,imgpi,
      BLAB1,LAB1,preimgpi,BJL,n,LS,SCTLStar1,
      h1,h2,LTLJ,BLTLJ,k1,hh,k2,kt,zz1,zz2,zz3,zz4,vls1,vls2,
      b1,b2,b3,b4,b5,o1,o2,ts,r,g1,bL1,kk1,ZStarL;


lisabelian:=0;
if IsLieAbelian(L) then lisabelian:=1; fi; 
K:=L!.LeftActingDomain;
BasisL:=Basis(L);
lenBL:=Length(BasisL);
SCTL:=StructureConstantsTable(BasisL); 
SCTC:=EmptySCTable(lenBL^2,0*One(K),"antisymmetric");

for u1 in [1..lenBL] do 
 for u2 in [1..lenBL] do
  for u3 in [1..lenBL] do
   for u4 in [1..lenBL] do
     l1:=Length(SCTL[u1][u2][1]);
     l2:=Length(SCTL[u3][u4][1]);
     if l1<>0 then if l2<>0 then 
                         index1:=(u1-1)*lenBL+u2;
                         index2:=(u3-1)*lenBL+u4;
                         derayeh1:=[];
 
                         for t1 in [1..l1] do
                          for t2 in [1..l2] do
                            i:=SCTL[u1][u2][1][t1];
                            j:=SCTL[u3][u4][1][t2];
                            m:=(i-1)*lenBL+j;
                            p:=SCTL[u1][u2][2][t1]*SCTL[u3][u4][2][t2];
                            Add(derayeh1,p*One(K));
                            Add(derayeh1,m);
                          od;
                         od;  
        
                         SetEntrySCTable(SCTC,index1,index2,derayeh1);
     fi; fi;
   od;
  od;
 od;
od;

C:=AlgebraByStructureConstants(K,SCTC);

vectorsI:=[];

for u1 in [1..lenBL] do 
 for u2 in [1..lenBL] do
  for u3 in [1..lenBL] do
    zr1:=[];
    mr1:=[];
    zr2:=[];
    mr2:=[];
    lenv1:=Length(SCTL[u1][u2][1]);
    if lenv1<>0 then
      for t1 in [1..lenv1] do
        i:=SCTL[u1][u2][1][t1];
        j:=u3;
        m:=(i-1)*lenBL+j;  
        p:=SCTL[u1][u2][2][t1]*1;
        Add(zr1,p*One(K)); 
        Add(mr1,m);
      od;
    fi;
    lenv2:=Length(SCTL[u2][u3][1]);
    if lenv2<>0 then
      for t2 in [1..lenv2] do
        i:=u1;
        j:=SCTL[u2][u3][1][t2];
        m:=(i-1)*lenBL+j;  
        p:=1*SCTL[u2][u3][2][t2];
        e:=0;
        if Length(zr1)<>0 then 
          for q in [1..Length(zr1)] do
            if mr1[q]=m then zr1[q]:=zr1[q]-p*One(K); e:=1; fi;
          od;      
          if e=0 then Add(zr1,-1*p*One(K)); Add(mr1,m); fi;
        fi;
        if Length(zr1)=0 then  Add(zr1,-1*p*One(K)); Add(mr1,m); fi;
      od;
    fi;
    lenv3:=Length(SCTL[u1][u3][1]);
    if lenv3<>0 then
      for t3 in [1..lenv3] do
        i:=u2;
        j:=SCTL[u1][u3][1][t3];
        m:=(i-1)*lenBL+j;  
        p:=1*SCTL[u1][u3][2][t3];
        e:=0; 
        if Length(zr1)<>0 then 
          for q in [1..Length(zr1)] do
            if mr1[q]=m then zr1[q]:=zr1[q]+p*One(K); e:=1; fi;
          od;
          if e=0 then Add(zr1,p*One(K)); Add(mr1,m); fi;
        fi;
        if Length(zr1)=0 then Add(zr1,p*One(K)); Add(mr1,m); fi;
      od;
    fi;
    lenv4:=Length(SCTL[u2][u3][1]);
    if lenv4<>0 then
      for t4 in [1..lenv2] do
        i:=u1;
        j:=SCTL[u2][u3][1][t4];
        m:=(i-1)*lenBL+j;  
        p:=1*SCTL[u2][u3][2][t4];
        Add(zr2,p*One(K)); 
        Add(mr2,m);
      od;
    fi;
    lenv5:=Length(SCTL[u3][u1][1]);
    if lenv5<>0 then
      for t5 in [1..lenv5] do
        i:=SCTL[u3][u1][1][t5];
        j:=u2; 
        m:=(i-1)*lenBL+j;  
        p:=SCTL[u3][u1][2][t5]*1;
        e:=0;
        if Length(zr2)<>0 then 
          for q in [1..Length(zr2)] do
            if mr2[q]=m then zr2[q]:=zr2[q]-p*One(K); e:=1; fi;
          od;      
          if e=0 then Add(zr2,-1*p*One(K)); Add(mr2,m); fi;
        fi;
        if Length(zr2)=0 then  Add(zr2,-1*p*One(K)); Add(mr2,m); fi;
      od;
    fi;
    lenv6:=Length(SCTL[u2][u1][1]);
    if lenv6<>0 then
      for t6 in [1..lenv6] do
        i:=SCTL[u2][u1][1][t6];
        j:=u3;
        m:=(i-1)*lenBL+j;  
        p:=SCTL[u2][u1][2][t6]*1;
        e:=0; 
        if Length(zr2)<>0 then 
          for q in [1..Length(zr2)] do
            if mr2[q]=m then zr2[q]:=zr2[q]+p*One(K); e:=1; fi;
          od;
          if e=0 then Add(zr2,p*One(K)); Add(mr2,m); fi;
        fi;
        if Length(zr2)=0 then Add(zr2,p*One(K)); Add(mr2,m); fi;
      od;
    fi;

    lzr1:=Length(zr1);
    lmr1:=Length(mr1);
    v1:=0;
    if lzr1<>0 then 
      v1:=zr1[1]*(Elements(Basis(C))[lenBL^2-mr1[1]+1]);
      for q1 in [2..lzr1] do
        if zr1[q1]<>0 then  
          v1:=v1+zr1[q1]*(Elements(Basis(C))[lenBL^2-mr1[q1]+1]);
        fi;
      od;
    fi;
    tt1:=0;
    for ww in [1..Length(Basis(C))] do
      if v1=0*Elements(Basis(C))[ww] then tt1:=1; fi;
    od; 
    w:=0; 
    for h in [1..Length(vectorsI)] do
      if v1=vectorsI[h] then w:=1; fi;
      if -1*v1=vectorsI[h] then w:=1; fi;
    od;
    if v1<>0 then if w=0 then if tt1=0 then Add(vectorsI,v1); fi; fi; fi;
    lzr2:=Length(zr2);
    lmr2:=Length(mr2);
    v2:=0;
    if lzr2<>0 then 
      v2:=zr2[1]*(Elements(Basis(C))[lenBL^2-mr2[1]+1]);
      for q2 in [2..lzr2] do
        if zr2[q2]<>0 then  
          v2:=v2+zr2[q2]*(Elements(Basis(C))[lenBL^2-mr2[q2]+1]);
        fi;
      od;
    fi;
    tt2:=0;
    for ww in [1..Length(Basis(C))] do
      if v2=0*Elements(Basis(C))[ww] then tt2:=1; fi;
    od; 
    w:=0; 
    for h in [1..Length(vectorsI)] do
      if v2=vectorsI[h] then w:=1; fi;
      if -1*v2=vectorsI[h] then w:=1; fi;
    od;
    if v2<>0 then if w=0 then if tt2=0 then Add(vectorsI,v2); fi; fi; fi;

  od;
 od;
od;

I:=Ideal(C,vectorsI);
LTL:=C/I;


MLTL:=[];

for i in [1..Length(Basis(C))] do
  v:=Elements(Basis(C))[lenBL^2-i+1];
  if not (v in I) then Add(MLTL,v); fi;
od;

BI:=Basis(I);
LenBI:=Length(BI);
BLTL:=[];
BBLTL:=[];

for j in [1..LenBI] do
  Add(BBLTL,Elements(BI)[j]);
od;

for i in [1..Length(MLTL)] do
  Add(BLTL,MLTL[i]);
  Add(BBLTL,MLTL[i]);
  I1:=VectorSpace(K,BBLTL);
  if Dimension(I1)=LenBI+1 then LenBI:=LenBI+1;  
    else 
      Remove(BLTL);
      Remove(BBLTL);
  fi;
  if Length(BLTL)=Dimension(C) then break; fi;
od;

I2:=VectorSpace(K,BLTL);
I3:=VectorSpace(K,Basis(LTL));
liltl:=[];

for i in [1..Length(BLTL)] do
 for j in [1..Length(Basis(C))] do
   if BLTL[i]=Elements(Basis(C))[lenBL^2-j+1] then 
     Add(liltl,j);         
     break;
   fi;
 od;
od; 

bL:=[];

for k in [1..Length(liltl)] do
  u1:=Int(liltl[k]/lenBL)+1;  
  u2:= liltl[k] mod lenBL;
  if u2=0 then u2:=lenBL; u1:=u1-1;  fi;
  u3:=SCTL[u1][u2];
  llzr1:=Length(SCTL[u1][u2][1]);
  vv1:=0*Elements(BasisL)[1];
  if llzr1<>0 then 
    vv1:=SCTL[u1][u2][2][1]*Elements(BasisL)[lenBL-SCTL[u1][u2][1][1]+1];
    for i in [2..llzr1] do
      vv1:=vv1+SCTL[u1][u2][2][i]*Elements(BasisL)[lenBL-SCTL[u1][u2][1][i]+1];
    od;
  fi;
  Add(bL,vv1);
od;

bLTL:=Basis(LTL);

g:= AlgebraHomomorphismByImages( LTL, L, bLTL , bL );;



LL:=LieDerivedSubalgebra(L);
f:= AlgebraHomomorphismByImages( LTL, LL, bLTL , bL );

L2:=Image(f);
LL:=LieDerivedSubalgebra(L);
dlab:=0;
nn:=Dimension(L)-Dimension(LL);
if nn=0 then dlab:=1; fi; 
Tens:=Source(f);
pif:=[];
BL2:=Basis(L2);
l:=Length(BL2);
if l<>0 then 
  for i in [1..l] do
    v:=Random(PreImagesElm(f,Basis(LL)[i]));
    Add(pif,v);
  od;
  vpif:=VectorSpace(K,pif);
  Bvpif:=Basis(vpif);
  lenbvpif:=Length(Bvpif);
  KK:=Elements(Bvpif);

  sp:=LeftModuleHomomorphismByImages(LL,LTL,Basis(LL),pif);

  imagesp:=Image(sp);
  bimagesp:=Basis(imagesp);
fi;

pi:= NaturalHomomorphismByIdeal( L, LL ); 

imgpi:=Image(pi);
BLAB1:=Basis(imgpi);
LAB1:=VectorSpace(K,BLAB1);
preimgpi:=[];
BJL:=[];
l:=Length(BLAB1);
if l<>0 then 
  for i in [1..l] do
    v:=Random(PreImagesElm(pi,BLAB1[i]));
    Add(preimgpi,v);
  od;
fi;
MLAB:=[];

for i in [1..Length(Basis(L))] do
  v:=Elements(Basis(L))[i];
  if not (v in L2) then Add(MLAB,v); fi;
od;

LenBL2:=Length(BL2);
BLAB:=[];
BBLAB:=[];

for j in [1..LenBL2] do
  Add(BBLAB,Elements(BL2)[j]);
od;

for i in [1..Length(MLAB)] do
  Add(BLAB,MLAB[i]);
  Add(BBLAB,MLAB[i]);
  IIII:=VectorSpace(K,BBLAB);
  if Dimension(IIII)=LenBL2+1 then LenBL2:=LenBL2+1;  
    else 
      Remove(BLAB);
      Remove(BBLAB);
  fi;
  if Length(BBLAB)=Dimension(L) then break; fi;
od;

LAB:=VectorSpace(K,BLAB);
l:=Length(BLAB);
if l<>0 then 
  for i in [1..l] do
    Add(BJL,BLAB[i]);
  od;
fi;
v1:=[];
v2:=[];
l:=Length(BL2);
if l<>0 then 
  for i in [1..Length(bLTL)] do
    v:=Image(f,bLTL[i]);
    h1:=1;
    if p<>0 then
      for r in [1..l] do
        if v=0*BL2[r] then h1:=0; fi;
      od;
      if h1=1 then 
        Add(BJL,v); 
        Add(v1,v);
        Add(v2,i); 
      fi;
    fi;
  od;
fi;
if Length(BLAB1)>0 then
  t:=LeftModuleHomomorphismByImages(LAB1,L,BLAB1,preimgpi);
fi;
m:=Dimension(LTL);
n:=Length(BLAB1);
p:=Length(BL2);

SCTLStar1:=EmptySCTable(m+n,0*One(K),"antisymmetric");

h:=NaturalHomomorphismByIdeal(C,I);;

LTLJ:=Image(h);
BLTLJ:=Basis(LTLJ);

for i in [1..n+m] do
 for j in [i..n+m] do
   if i<=m then
     k1:=Image(f,bLTL[i]);
     h1:=1;
     if p<>0 then
       for r in [1..p] do
         if k1=0*Basis(L2)[r] then h1:=0; fi;
       od;
       if h1=1 then 
         for hh in [1..Length(v1)] do
           if v1[hh]=k1 then k1:=BJL[n+hh]; fi;
         od; 
       fi;

       else
         h1:=0;
     fi; 
 
    else
      k1:=BJL[i-m];
      h1:=1;
  fi;

  if j<=m then
    k2:=Image(f,bLTL[j]);
    h2:=1;
    if p<>0 then
      for r in [1..p] do
        if k2=0*Basis(L2)[r] then h2:=0; fi;
      od;
      if h2=1 then 
        for hh in [1..Length(v1)] do
          if v1[hh]=k2 then k2:=BJL[n+hh]; fi;
        od; 
      fi;

      else
        h2:=0;
        fi;

      else
        k2:=BJL[j-m];
        h2:=1;
   fi;

   if h1=1 and h2=1 then
     zz1:=Coefficients(BasisL, k1 );
     zz2:=Coefficients(BasisL, k2 );
       vls1:=[];
       for b4 in [1..m] do
         Add(vls1,0);
       od;

       for b1 in [1..Length(zz1)] do
        for b2 in [1..Length(zz2)] do
          o1:=(b1-1)*lenBL+b2;
          o2:=zz1[b1]*zz2[b2];
          if o2<>0 then   
            kt:=Image(h,Basis(C)[o1]);
            zz3:=Coefficients(BLTLJ,kt);
              for b3 in [1..m] do
                vls1[b3]:=vls1[b3]+o2*zz3[b3];
              od;            
           fi;
        od;
       od; 

   vls2:=[];
   for b5 in [1..m] do
     if vls1[b5]<>0 then
       Add(vls2,vls1[b5]);
       Add(vls2,b5); 
     fi;
   od;
   if Length(vls2)>0 then SetEntrySCTable( SCTLStar1, i, j, vls2 );  fi;
 fi;

 od;
od;
LS:=LieAlgebraByStructureConstants(K,SCTLStar1);

bL1:=[];

for i in [1..Length(bL)] do 
 Add(bL1,bL[i]);
od;

for i in [1..n] do
 Add(bL1,BJL[i]);
od;


#bL1:=[];
#for i in [1..Length(BasisL)] do
#  Add(bL1,BasisL[i]);
#od;

#for i in [1..(Length(bLTL)-p)] do
#  Add(bL1,0*BasisL[1]);
#od;

g1:= AlgebraHomomorphismByImages( LS, L, Basis(LS) , bL1 );

return g1;
end);