Spracherkennung für: .gi vermutete Sprache: Unknown {[0] [0] [0]} [Methode: Schwerpunktbildung, einfache Gewichte, sechs Dimensionen]
InstallMethod( SolvableLieAlgebra, "for a field and a list", true,
[ IsField, IsList ], 0,
function( F, data )
local d, no, a, b, S, L;
if not Length( data ) >= 2 then
Error("<data> has to have length at least two");
fi;
if not IsPosInt( data[1] ) then
Error("the first element of <data> has to be a positive integer");
else
d:= data[1];
if not d in [1,2,3,4] then
Error("the dimension has to be 1,2,3, or 4");
fi;
fi;
no:= data[2];
if d=1 then
if no > 1 then
Error( "the second element of <data> has to be <= 1" );
fi;
elif d=2 then
if no > 2 then
Error("the second element of <data> has to be <= 2");
fi;
elif d=3 then
if no > 4 then
Error("the second element of <data> has to be <= 4");
fi;
else
if not no in [1..14] then
Error("the second element of <data> has to be <= 14");
fi;
fi;
if Length( data ) >=3 then
a:= data[3];
fi;
if Length( data ) >= 4 then
b:= data[4];
fi;
if d = 1 then
S:= EmptySCTable( 1, Zero(F), "antisymmetric" );
L := LieAlgebraByStructureConstants( F, S );
L!.arg := data;
return L;
elif d = 2 then
if no=1 then
S:= EmptySCTable( 2, Zero(F), "antisymmetric" );
L := LieAlgebraByStructureConstants( F, S );
L!.arg := data;
return L;
else
S:= EmptySCTable( 2, Zero(F), "antisymmetric" );
SetEntrySCTable( S, 2, 1, [One(F),1] );
L := LieAlgebraByStructureConstants( F, S );
L!.arg := data;
return L;
fi;
elif d=3 then
if no=1 then
S:= EmptySCTable( 3, Zero(F), "antisymmetric" );
L := LieAlgebraByStructureConstants( F, S );
L!.arg := data;
return L;
elif no=2 then
S:= EmptySCTable( 3, Zero(F), "antisymmetric" );
SetEntrySCTable( S, 3, 1, [One(F),1] );
SetEntrySCTable( S, 3, 2, [One(F),2] );
L := LieAlgebraByStructureConstants( F, S );
L!.arg := data;
return L;
elif no=3 then
S:= EmptySCTable( 3, Zero(F), "antisymmetric" );
SetEntrySCTable( S, 3, 1, [One(F),2] );
SetEntrySCTable( S, 3, 2, [a,1,One(F),2] );
L := LieAlgebraByStructureConstants( F, S );
L!.arg := data;
return L;
else
S:= EmptySCTable( 3, Zero(F), "antisymmetric" );
SetEntrySCTable( S, 3, 1, [One(F),2] );
SetEntrySCTable( S, 3, 2, [a,1] );
L := LieAlgebraByStructureConstants( F, S );
L!.arg := data;
return L;
fi;
else
if no=1 then
S:= EmptySCTable( 4, Zero(F), "antisymmetric" );
L := LieAlgebraByStructureConstants( F, S );
L!.arg := data;
return L;
elif no=2 then
S:= EmptySCTable( 4, Zero(F), "antisymmetric" );
SetEntrySCTable( S, 4, 1, [One(F),1] );
SetEntrySCTable( S, 4, 2, [One(F),2] );
SetEntrySCTable( S, 4, 3, [One(F),3] );
L := LieAlgebraByStructureConstants( F, S );
L!.arg := data;
return L;
elif no=3 then
S:= EmptySCTable( 4, Zero(F), "antisymmetric" );
SetEntrySCTable( S, 4, 1, [One(F),1] );
SetEntrySCTable( S, 4, 2, [One(F),3] );
SetEntrySCTable( S, 4, 3, [-a,2,a+One(F),3] );
L := LieAlgebraByStructureConstants( F, S );
L!.arg := data;
return L;
elif no = 4 then
S:= EmptySCTable( 4, Zero(F), "antisymmetric" );
SetEntrySCTable( S, 4, 2, [One(F),3] );
SetEntrySCTable( S, 4, 3, [One(F),3] );
L := LieAlgebraByStructureConstants( F, S );
L!.arg := data;
return L;
elif no = 5 then
S:= EmptySCTable( 4, Zero(F), "antisymmetric" );
SetEntrySCTable( S, 4, 2, [One(F),3] );
L := LieAlgebraByStructureConstants( F, S );
L!.arg := data;
return L;
elif no = 6 then
S:= EmptySCTable( 4, Zero(F), "antisymmetric" );
SetEntrySCTable( S, 4, 1, [One(F),2] );
SetEntrySCTable( S, 4, 2, [One(F),3] );
SetEntrySCTable( S, 4, 3, [a,1,b,2,One(F),3] );
L := LieAlgebraByStructureConstants( F, S );
L!.arg := data;
return L;
elif no = 7 then
S:= EmptySCTable( 4, Zero(F), "antisymmetric" );
SetEntrySCTable( S, 4, 1, [One(F),2] );
SetEntrySCTable( S, 4, 2, [One(F),3] );
SetEntrySCTable( S, 4, 3, [a,1,b,2] );
L := LieAlgebraByStructureConstants( F, S );
L!.arg := data;
return L;
elif no=8 then
S:= EmptySCTable( 4, Zero(F), "antisymmetric" );
SetEntrySCTable( S, 1, 2, [One(F),2] );
SetEntrySCTable( S, 3, 4, [One(F),4] );
L := LieAlgebraByStructureConstants( F, S );
L!.arg := data;
return L;
elif no=9 then
S:= EmptySCTable( 4, Zero(F), "antisymmetric" );
SetEntrySCTable( S, 4, 1, [One(F),1,a,2] );
SetEntrySCTable( S, 4, 2, [One(F),1] );
SetEntrySCTable( S, 3, 1, [One(F),1] );
SetEntrySCTable( S, 3, 2, [One(F),2] );
L := LieAlgebraByStructureConstants( F, S );
L!.arg := data;
return L;
elif no=10 then
S:= EmptySCTable( 4, Zero(F), "antisymmetric" );
SetEntrySCTable( S, 4, 1, [One(F),2] );
SetEntrySCTable( S, 4, 2, [a,1] );
SetEntrySCTable( S, 3, 1, [One(F),1] );
SetEntrySCTable( S, 3, 2, [One(F),2] );
L := LieAlgebraByStructureConstants( F, S );
L!.arg := data;
return L;
elif no=11 then
if Characteristic(F) <> 2 then
Error( "The characteristic of F has to be 2 for L4_11");
fi;
S:= EmptySCTable( 4, Zero(F), "antisymmetric" );
SetEntrySCTable( S, 4, 1, [One(F),1] );
SetEntrySCTable( S, 4, 2, [b,2] );
SetEntrySCTable( S, 4, 3, [One(F)+b,3] );
SetEntrySCTable( S, 3, 1, [One(F),2] );
SetEntrySCTable( S, 3, 2, [a,1] );
L := LieAlgebraByStructureConstants( F, S );
L!.arg := data;
return L;
elif no=12 then
S:= EmptySCTable( 4, Zero(F), "antisymmetric" );
SetEntrySCTable( S, 4, 1, [One(F),1] );
SetEntrySCTable( S, 4, 2, [2*One(F),2] );
SetEntrySCTable( S, 4, 3, [One(F),3] );
SetEntrySCTable( S, 3, 1, [One(F),2] );
L := LieAlgebraByStructureConstants( F, S );
L!.arg := data;
return L;
elif no=13 then
S:= EmptySCTable( 4, Zero(F), "antisymmetric" );
SetEntrySCTable( S, 4, 1, [One(F),1,a,3] );
SetEntrySCTable( S, 4, 2, [One(F),2] );
SetEntrySCTable( S, 4, 3, [One(F),1] );
SetEntrySCTable( S, 3, 1, [One(F),2] );
L := LieAlgebraByStructureConstants( F, S );
L!.arg := data;
return L;
else
S:= EmptySCTable( 4, Zero(F), "antisymmetric" );
SetEntrySCTable( S, 4, 1, [a,3] );
SetEntrySCTable( S, 4, 3, [One(F),1] );
SetEntrySCTable( S, 3, 1, [One(F),2] );
L := LieAlgebraByStructureConstants( F, S );
L!.arg := data;
return L;
fi;
fi;
end );
InstallMethod( NilpotentLieAlgebra, "for a field and a list", true,
[ IsField, IsList ], 0,
function( F, data )
local L, d, no, a, ff, arg,
N1_1, N2_1, N3_1, N3_2, N4_1, N4_2, N4_3, N5_1, N5_2, N5_3,
N5_4, N5_5, N5_6, N5_7, N5_8, N5_9, N6_1, N6_2, N6_3, N6_4,
N6_5, N6_6, N6_7, N6_8, N6_9, N6_10, N6_11, N6_12, N6_13, N6_14,
N6_15, N6_16, N6_17, N6_18, N6_19, N6_20, N6_21, N6_22, N6_23, N6_24,
N6_25, N6_26, N6_27, N6_28, N6_29, N6_30, N6_31, N6_32, N6_33, N6_34,
N6_35, N6_36 ;
N1_1 := function( F )
local T;
T := EmptySCTable( 1, Zero(F), "antisymmetric" );
return LieAlgebraByStructureConstants( F, T );
end;
N2_1 := function( F )
local T;
T := EmptySCTable( 2, Zero(F), "antisymmetric" );
return LieAlgebraByStructureConstants( F, T );
end;
N3_1 := function( F )
local T;
T := EmptySCTable( 3, Zero(F), "antisymmetric" );
return LieAlgebraByStructureConstants( F, T );
end;
N3_2 := function( F )
local T;
T := EmptySCTable( 3, Zero(F), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
return LieAlgebraByStructureConstants( F, T );
end;
N4_1 := function( F )
local T;
T := EmptySCTable( 4, Zero(F), "antisymmetric" );
return LieAlgebraByStructureConstants( F, T );
end;
N4_2 := function( F )
local T;
T := EmptySCTable( 4, Zero(F), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
return LieAlgebraByStructureConstants( F, T );
end;
N4_3 := function( F )
local T;
T := EmptySCTable( 4, Zero(F), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,4] );
return LieAlgebraByStructureConstants( F, T );
end;
N5_1:= function( F )
local T;
T:= EmptySCTable( 5, Zero(F), "antisymmetric" );
return LieAlgebraByStructureConstants( F, T );
end;
N5_2:= function( F )
local T;
T:= EmptySCTable( 5, Zero(F), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
return LieAlgebraByStructureConstants( F, T );
end;
N5_3:= function( F )
local T;
T:= EmptySCTable( 5, Zero(F), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,4] );
return LieAlgebraByStructureConstants( F, T );
end;
N5_4:= function( F )
local T;
T:= EmptySCTable( 5, Zero(F), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,5] );
SetEntrySCTable( T, 3, 4, [1,5] );
return LieAlgebraByStructureConstants( F, T );
end;
N5_5:= function( F )
local T;
T:= EmptySCTable( 5, Zero(F), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,5] );
SetEntrySCTable( T, 2, 4, [1,5] );
return LieAlgebraByStructureConstants( F, T );
end;
N5_6:= function( F )
local T;
T:= EmptySCTable( 5, Zero(F), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,4] );
SetEntrySCTable( T, 1, 4, [1,5] );
SetEntrySCTable( T, 2, 3, [1,5] );
return LieAlgebraByStructureConstants( F, T );
end;
N5_7:= function( F )
local T;
T:= EmptySCTable( 5, Zero(F), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,4] );
SetEntrySCTable( T, 1, 4, [1,5] );
return LieAlgebraByStructureConstants( F, T );
end;
N5_8:= function( F )
local T;
T:= EmptySCTable( 5, Zero(F), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,4] );
SetEntrySCTable( T, 1, 3, [1,5] );
return LieAlgebraByStructureConstants( F, T );
end;
N5_9:= function( F )
local T;
T:= EmptySCTable( 5, Zero(F), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,4] );
SetEntrySCTable( T, 2, 3, [1,5] );
return LieAlgebraByStructureConstants( F, T );
end;
N6_1:= function( F )
local T;
T:= EmptySCTable( 6, Zero(F), "antisymmetric" );
return LieAlgebraByStructureConstants( F, T );
end;
N6_2:= function( F )
local T;
T:= EmptySCTable( 6, Zero(F), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
return LieAlgebraByStructureConstants( F, T );
end;
N6_3:= function( F )
local T;
T:= EmptySCTable( 6, Zero(F), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,4] );
return LieAlgebraByStructureConstants( F, T );
end;
N6_4:= function( F )
local T;
T:= EmptySCTable( 6, Zero(F), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,5] );
SetEntrySCTable( T, 3, 4, [1,5] );
return LieAlgebraByStructureConstants( F, T );
end;
N6_5:= function( F )
local T;
T:= EmptySCTable( 6, Zero(F), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,5] );
SetEntrySCTable( T, 2, 4, [1,5] );
return LieAlgebraByStructureConstants( F, T );
end;
N6_6:= function( F )
local T;
T:= EmptySCTable( 6, Zero(F), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,4] );
SetEntrySCTable( T, 1, 4, [1,5] );
SetEntrySCTable( T, 2, 3, [1,5] );
return LieAlgebraByStructureConstants( F, T );
end;
N6_7:= function( F )
local T;
T:= EmptySCTable( 6, Zero(F), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,4] );
SetEntrySCTable( T, 1, 4, [1,5] );
return LieAlgebraByStructureConstants( F, T );
end;
N6_8:= function( F )
local T;
T:= EmptySCTable( 6, Zero(F), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,4] );
SetEntrySCTable( T, 1, 3, [1,5] );
return LieAlgebraByStructureConstants( F, T );
end;
N6_9:= function( F )
local T;
T:= EmptySCTable( 6, Zero(F), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,4] );
SetEntrySCTable( T, 2, 3, [1,5] );
return LieAlgebraByStructureConstants( F, T );
end;
N6_10:= function( F )
local T;
T:= EmptySCTable( 6, Zero(F), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,6] );
SetEntrySCTable( T, 4, 5, [1,6] );
return LieAlgebraByStructureConstants( F, T );
end;
N6_11:= function( F )
local T;
T:= EmptySCTable( 6, Zero(F), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,4] );
SetEntrySCTable( T, 1, 4, [1,6] );
SetEntrySCTable( T, 2, 3, [1,6] );
SetEntrySCTable( T, 2, 5, [1,6] );
return LieAlgebraByStructureConstants( F, T );
end;
N6_12:= function( F )
local T;
T:= EmptySCTable( 6, Zero(F), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,4] );
SetEntrySCTable( T, 1, 4, [1,6] );
SetEntrySCTable( T, 2, 5, [1,6] );
return LieAlgebraByStructureConstants( F, T );
end;
N6_13:= function( F )
local T;
T:= EmptySCTable( 6, Zero(F), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,5] );
SetEntrySCTable( T, 1, 5, [1,6] );
SetEntrySCTable( T, 2, 4, [1,5] );
SetEntrySCTable( T, 3, 4, [1,6] );
return LieAlgebraByStructureConstants( F, T );
end;
N6_14:= function( F )
local T;
T:= EmptySCTable( 6, Zero(F), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,4] );
SetEntrySCTable( T, 1, 4, [1,5] );
SetEntrySCTable( T, 2, 3, [1,5] );
SetEntrySCTable( T, 2, 5, [1,6] );
SetEntrySCTable( T, 3, 4, [-1,6] );
return LieAlgebraByStructureConstants( F, T );
end;
N6_15:= function( F )
local T;
T:= EmptySCTable( 6, Zero(F), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,4] );
SetEntrySCTable( T, 1, 4, [1,5] );
SetEntrySCTable( T, 1, 5, [1,6] );
SetEntrySCTable( T, 2, 3, [1,5] );
SetEntrySCTable( T, 2, 4, [1,6] );
return LieAlgebraByStructureConstants( F, T );
end;
N6_16:= function( F )
local T;
T:= EmptySCTable( 6, Zero(F), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,4] );
SetEntrySCTable( T, 1, 4, [1,5] );
SetEntrySCTable( T, 2, 5, [1,6] );
SetEntrySCTable( T, 3, 4, [-1,6] );
return LieAlgebraByStructureConstants( F, T );
end;
N6_17:= function( F )
local T;
T:= EmptySCTable( 6, Zero(F), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,4] );
SetEntrySCTable( T, 1, 4, [1,5] );
SetEntrySCTable( T, 1, 5, [1,6] );
SetEntrySCTable( T, 2, 3, [1,6] );
return LieAlgebraByStructureConstants( F, T );
end;
N6_18:= function( F )
local T;
T:= EmptySCTable( 6, Zero(F), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,4] );
SetEntrySCTable( T, 1, 4, [1,5] );
SetEntrySCTable( T, 1, 5, [1,6] );
return LieAlgebraByStructureConstants( F, T );
end;
N6_19:= function( F, a )
local T;
T:= EmptySCTable( 6, Zero(F), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,4] );
SetEntrySCTable( T, 1, 3, [1,5] );
SetEntrySCTable( T, 1, 5, [1,6] );
SetEntrySCTable( T, 2, 4, [1,6] );
SetEntrySCTable( T, 3, 5, [a,6] );
return LieAlgebraByStructureConstants( F, T );
end;
N6_20:= function( F )
local T;
T:= EmptySCTable( 6, Zero(F), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,4] );
SetEntrySCTable( T, 1, 3, [1,5] );
SetEntrySCTable( T, 1, 5, [1,6] );
SetEntrySCTable( T, 2, 4, [1,6] );
return LieAlgebraByStructureConstants( F, T );
end;
N6_21:= function( F, a )
local T;
T:= EmptySCTable( 6, Zero(F), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,4] );
SetEntrySCTable( T, 1, 4, [1,6] );
SetEntrySCTable( T, 2, 3, [1,5] );
SetEntrySCTable( T, 2, 5, [a,6] );
return LieAlgebraByStructureConstants( F, T );
end;
N6_22:= function( F, a )
local T;
T:= EmptySCTable( 6, Zero(F), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,5] );
SetEntrySCTable( T, 1, 3, [1,6] );
SetEntrySCTable( T, 2, 4, [a,6] );
SetEntrySCTable( T, 3, 4, [1,5] );
return LieAlgebraByStructureConstants( F, T );
end;
N6_23:= function( F )
local T;
T:= EmptySCTable( 6, Zero(F), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,5] );
SetEntrySCTable( T, 1, 4, [1,6] );
SetEntrySCTable( T, 2, 4, [1,5] );
return LieAlgebraByStructureConstants( F, T );
end;
N6_24:= function( F, a )
local T;
T:= EmptySCTable( 6, Zero(F), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,5] );
SetEntrySCTable( T, 1, 4, [a,6] );
SetEntrySCTable( T, 2, 3, [1,6] );
SetEntrySCTable( T, 2, 4, [1,5] );
return LieAlgebraByStructureConstants( F, T );
end;
N6_25:= function( F )
local T;
T:= EmptySCTable( 6, Zero(F), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,5] );
SetEntrySCTable( T, 1, 4, [1,6] );
return LieAlgebraByStructureConstants( F, T );
end;
N6_26:= function( F )
local T;
T:= EmptySCTable( 6, Zero(F), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,4] );
SetEntrySCTable( T, 1, 3, [1,5] );
SetEntrySCTable( T, 2, 3, [1,6] );
return LieAlgebraByStructureConstants( F, T );
end;
N6_27:= function( F )
local T;
T:= EmptySCTable( 6, Zero(F), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,5] );
SetEntrySCTable( T, 2, 4, [1,6] );
return LieAlgebraByStructureConstants( F, T );
end;
N6_28:= function( F )
local T;
T:= EmptySCTable( 6, Zero(F), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,4] );
SetEntrySCTable( T, 1, 4, [1,5] );
SetEntrySCTable( T, 2, 3, [1,6] );
return LieAlgebraByStructureConstants( F, T );
end;
N6_29:= function( F )
local T;
T:= EmptySCTable( 6, Zero(F), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,5] );
SetEntrySCTable( T, 1, 5, [1,6] );
SetEntrySCTable( T, 2, 4, [1,5,1,6] );
SetEntrySCTable( T, 3, 4, [1,6] );
return LieAlgebraByStructureConstants( F, T );
end;
N6_30:= function( F )
local T;
T:= EmptySCTable( 6, Zero(F), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,4] );
SetEntrySCTable( T, 1, 4, [1,5] );
SetEntrySCTable( T, 1, 5, [1,6] );
SetEntrySCTable( T, 2, 3, [1,5,1,6] );
SetEntrySCTable( T, 2, 4, [1,6] );
return LieAlgebraByStructureConstants( F, T );
end;
N6_31:= function( F, a )
local T;
T:= EmptySCTable( 6, Zero(F), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,4] );
SetEntrySCTable( T, 1, 4, [1,5] );
SetEntrySCTable( T, 2, 3, [1,5,a,6] );
SetEntrySCTable( T, 2, 5, [1,6] );
SetEntrySCTable( T, 3, 4, [1,6] );
return LieAlgebraByStructureConstants( F, T );
end;
N6_32:= function( F, a )
local T;
T:= EmptySCTable( 6, Zero(F), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,4] );
SetEntrySCTable( T, 1, 4, [1,5] );
SetEntrySCTable( T, 2, 3, [a,6] );
SetEntrySCTable( T, 2, 5, [1,6] );
SetEntrySCTable( T, 3, 4, [1,6] );
return LieAlgebraByStructureConstants( F, T );
end;
N6_33:= function( F )
local T;
T:= EmptySCTable( 6, Zero(F), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,4] );
SetEntrySCTable( T, 1, 3, [1,5] );
SetEntrySCTable( T, 2, 5, [1,6] );
SetEntrySCTable( T, 3, 4, [1,6] );
return LieAlgebraByStructureConstants( F, T );
end;
N6_34:= function( F )
local T;
T:= EmptySCTable( 6, Zero(F), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,4] );
SetEntrySCTable( T, 1, 5, [1,6] );
SetEntrySCTable( T, 2, 3, [1,5] );
SetEntrySCTable( T, 2, 4, [1,6] );
return LieAlgebraByStructureConstants( F, T );
end;
N6_35:= function( F, a )
local T;
T:= EmptySCTable( 6, Zero(F), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,5] );
SetEntrySCTable( T, 1, 3, [1,6] );
SetEntrySCTable( T, 2, 4, [a,6] );
SetEntrySCTable( T, 3, 4, [1,5,1,6] );
return LieAlgebraByStructureConstants( F, T );
end;
N6_36:= function( F, a )
local T;
T:= EmptySCTable( 6, Zero(F), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,5] );
SetEntrySCTable( T, 1, 4, [a,6] );
SetEntrySCTable( T, 2, 3, [1,6] );
SetEntrySCTable( T, 2, 4, [1,5,1,6] );
return LieAlgebraByStructureConstants( F, T );
end;
if not Length( data ) >= 2 then
Error("<data> has to have length at least two");
fi;
if not IsPosInt( data[1] ) then
Error("the first element of <data> has to be a positive integer");
else
d:= data[1];
if not d in [1,2,3,4,5,6] then
Error("the dimension has to be an integer from 1 to 6");
fi;
fi;
no:= data[2];
if d=1 then
if no > 1 then
Error( "the second element of <data> has to be <= 1" );
fi;
elif d=2 then
if no > 1 then
Error( "the second element of <data> has to be <= 1" );
fi;
elif d=3 then
if no > 2 then
Error( "the second element of <data> has to be <= 2" );
fi;
elif d=4 then
if no > 3 then
Error( "the second element of <data> has to be <= 3" );
fi;
elif d=5 then
if no > 9 then
Error("the second element of <data> has to be <= 9");
fi;
else
if no > 36 then
Error("the second element of <data> has to be <= 36");
fi;
fi;
if Length( data ) >= 3 then a:= data[3]; fi;
if d = 1 then
ff := [N1_1];
L := CallFuncList( ff[no], [ F ] );
L!.arg := data;
return L;
elif d = 2 then
ff := [N2_1];
L := CallFuncList( ff[no], [ F ] );
L!.arg := data;
return L;
elif d = 3 then
ff := [N3_1,N3_2];
L := CallFuncList( ff[no], [ F ] );
L!.arg := data;
return L;
elif d = 4 then
ff := [N4_1,N4_2,N4_3];
L := CallFuncList( ff[no], [ F ] );
L!.arg := data;
return L;
elif d = 5 then
ff:= [ N5_1, N5_2, N5_3, N5_4, N5_5, N5_6, N5_7, N5_8, N5_9 ];
L := CallFuncList( ff[no], [ F ] );
L!.arg := data;
return L;
else
ff:= [ N6_1, N6_2, N6_3, N6_4, N6_5, N6_6, N6_7, N6_8, N6_9, N6_10,
N6_11, N6_12, N6_13, N6_14, N6_15, N6_16, N6_17, N6_18, N6_19,
N6_20, N6_21, N6_22, N6_23, N6_24, N6_25, N6_26, N6_27, N6_28,
N6_29, N6_30, N6_31, N6_32, N6_33, N6_34, N6_35, N6_36];
if no in [19,21,22,24,31,32,35,36] then
arg:= [F,a];
else
arg:= [F];
fi;
L := CallFuncList( ff[no], arg );
L!.arg := data;
return L;
fi;
end );
BindGlobal("LieAlgDBHelper", rec());
LieAlgDBHelper.isomN:= function( L, x1, x2, x3, x4 )
# Here the xi satisfy the commutation relations of N;
# we produce the isomorphism with M_0^13...
local F, y1, y2, y3, y4, name, K, f;
F:= LeftActingDomain( L );
if Characteristic(F) <> 3 then
y1:= x1+x2;
y2:= 3*x1-(3/2*One(F))*x2;
y3:= x1+x2-x3+2*x4;
y4:= x3;
else
y1:= x2;
y2:= x1+x2;
y3:= x1-x2+x3+x4;
y4:= x1-x2+x3;
fi;
name:= "L4_13( ";
Append( name, LieAlgDBField2String( F ) );
Append( name, ", " );
Append( name, String( Zero(F) ) );
Append( name, " )" );
K:= SolvableLieAlgebra( F, [4,13, Zero(F)] );
f:= AlgebraHomomorphismByImages(K,L,Basis(K),[y1,y2,y3,y4]);
return rec( name:= [ name, [ Zero(F) ] ], isom:= f );
end;
LieAlgDBHelper.isomM9:= function( L, x1, x2, x3, x4, a )
# The xi satisfy the commutation rules of M9a; we return
# the isomorphism with either N, M8 or M9a
local F, T, pol, facs, dd, id, f, name, K, c, exp, prim, ef, b, q, i;
F:= LeftActingDomain( L );
if Characteristic(F) <> 2 and a = (-1/4)*One(F) then
return LieAlgDBHelper.isomN( L, x1, x2, x3, x4 );
fi;
T:= Indeterminate( F);
pol:= T^2-T-a;
facs:= Factors( pol );
if Length( facs ) = 2 then
#direct sum...
dd:= DirectSumDecomposition( L );
id:= LieAlgebraIdentification( dd[1] );
f:= id.isomorphism;
x1:= Image( f, Basis( Source(f) )[2] );
x2:= Image( f, Basis( Source(f) )[1] );
id:= LieAlgebraIdentification( dd[2] );
f:= id.isomorphism;
x3:= Image( f, Basis( Source(f) )[2] );
x4:= Image( f, Basis( Source(f) )[1] );
name:= "L4_8( ";
Append( name, LieAlgDBField2String( F ) );
Append( name, " )" );
K:= SolvableLieAlgebra( F, [4,8] );
f:= AlgebraHomomorphismByImages(K,L,Basis(K),[x1,x2,x3,x4]);
return rec( name:= [ name, [] ], isom:= f );
else
if HasIsFinite(F) and IsFinite(F) then
# for M9 we let b be the smallest power of the primitive root
# such that T^2-T-b has no roots in F
q:= Size( F );
prim:= PrimitiveRoot( F );
for i in [1..q-1] do
if Length(Factors( T^2-T-prim^i ) ) = 1 then
b:= prim^i;
break;
fi;
od;
if Characteristic(F) > 2 then
c:= (b+One(F)/4)/(a+One(F)/4);
exp:= LogFFE( c, PrimitiveRoot(F) );
c:= PrimitiveRoot(F)^(exp/2);
x1:= c*x1+((One(F)-c)/2)*x2;
x4:= ((One(F)-c)/2)*x3+c*x4;
else
facs:= Factors( T^2+T+a+b );
ef:= ExtRepPolynomialRatFun( facs[1] );
if Length( ef[1] ) = 0 then
c:= -ef[2];
else
c:= Zero( F );
fi;
x1:= x1+c*x2;
x4:= c*x3+x4;
fi;
else
b:= a;
fi;
name:= "L4_9( ";
Append( name, LieAlgDBField2String( F ) );
Append( name, ", " );
Append( name, String( b ) );
Append( name, " )" );
K:= SolvableLieAlgebra( F, [4,9, b] );
f:= AlgebraHomomorphismByImages(K,L,Basis(K),[x1,x2,x3,x4]);
return rec( name:= [ name, [ b ] ], isom:= f );
fi;
end;
LieAlgDBHelper.isomM10:= function( L, x1, x2, x3, x4, a )
# the xi satisfy the commutation rels of M10(a)
local f, b, y1, y2, y3, y4, name, K, F;
F:= LeftActingDomain( L );
if HasIsFinite(F) and IsFinite(F) then
f:= Inverse( FrobeniusAutomorphism(F) );
b:= Image( f, a );
y1:= x1;
y2:= b*x1+x2;
y3:= x3;
y4:= b*x3+x4;
name:= "L4_13( ";
Append( name, LieAlgDBField2String( F ) );
Append( name, ", " );
Append( name, String( Zero(F) ) );
Append( name, " )" );
K:= SolvableLieAlgebra( F, [4,13, Zero(F)] );
f:= AlgebraHomomorphismByImages(K,L,Basis(K),
[y1,y2,y1+y2+y4,y1+y2+y3]);
return rec( name:= [ name, [Zero(F)] ], isom:= f );
else
name:= "L4_10( ";
Append( name, LieAlgDBField2String( F ) );
Append( name, ", " );
Append( name, String( a ) );
Append( name, " )" );
K:= SolvableLieAlgebra( F, [4,10, a] );
f:= AlgebraHomomorphismByImages(K,L,Basis(K),[x1,x2,x3,x4]);
return rec( name:= [ name, [a] ], isom:= f );
fi;
end;
LieAlgDBHelper.isomM11:= function( L, x1, x2, x3, x4, a, b )
local name, K, f, gam, eps, del, y1, y2, y3, y4, F;
F:= LeftActingDomain(L);
if b = Zero(F) then
name:= "L4_11( ";
Append( name, LieAlgDBField2String( F ) );
Append( name, ", " );
Append( name, String( a ) );
Append( name, ", " );
Append( name, String( b ) );
Append( name, " )" );
K:= SolvableLieAlgebra( F, [4,11,a, b] );
f:= AlgebraHomomorphismByImages(K,L,Basis(K),[x1,x2,x3,x4]);
return rec( name:= [ name, [a,b] ], isom:= f );
elif HasIsFinite(F) and IsFinite(F) then
f:= Inverse( FrobeniusAutomorphism(F) );
gam:= Image( f, b/(a*(b^2+One(F))) );
eps:= Image( f, 1/a );
del:= 1/(1+b);
y1:= a*gam*x1+b*del*x2;
y2:= a*eps*b*del*x1+a*gam*eps*x2;
y3:= eps*x3;
y4:= gam*x3+del*x4;
name:= "L4_11( ";
Append( name, LieAlgDBField2String( F ) );
Append( name, ", " );
Append( name, String( One(F) ) );
Append( name, ", " );
Append( name, String( Zero(F) ) );
Append( name, " )" );
K:= SolvableLieAlgebra( F, [4,11, One(F), Zero(F)] );
f:= AlgebraHomomorphismByImages(K,L,Basis(K),[y1,y2,y3,y4]);
return rec( name:= [ name, [One(F),Zero(F)] ],isom:= f);
else
name:= "L4_11( ";
Append( name, LieAlgDBField2String( F ) );
Append( name, ", " );
Append( name, String( a ) );
Append( name, ", " );
Append( name, String( b ) );
Append( name, " )" );
K:= SolvableLieAlgebra( F, [4,11, a, b] );
f:= AlgebraHomomorphismByImages(K,L,Basis(K),[x1,x2,x3,x4]);
return rec( name:= [ name, [a,b] ], isom:= f );
fi;
end;
LieAlgDBHelper.solv_type:= function( L )
local n, F, sp, k, BL, b, x1, x2, x3, x4, c1, c2, K, f, name, par,
id, num, c3, c4, mat, is_diag, i, j, facs, ev, s, t, sol,
sevec, tevec, cfs, u, v, w, found, y1, y2, y3, y4, u1, u3,
v1, v3, a, D, adM, adK, dd, bK, b_adK, ef, ev1, ev2, mp, q, C,
R, exp;
n:= Dimension( L );
F:= LeftActingDomain( L );
sp:= LieDerivedSubalgebra( L );
k:= 1;
BL:= Basis( L );
while Dimension( sp ) < n-1 do
if not BL[k] in sp then
b:= ShallowCopy( BasisVectors( Basis(sp) ) );
Add( b, BL[k] );
sp:= Subspace( L, b );
fi;
k:= k+1;
od;
b:= Basis( sp );
if n = 2 then
x1:= b[1];
for k in [1,2] do
if not BL[k] in sp then
x2:= BL[k];
break;
fi;
od;
if x1*x2 = Zero(L) then
name:= "L2_1( ";
Append( name, LieAlgDBField2String( F ) );
Append( name, " )" );
K:= SolvableLieAlgebra( F, [2,1] );
f:= AlgebraHomomorphismByImages( K, L, Basis(K), Basis(L) );
return rec( name:= [ name, [] ], isom:= f );
fi;
c1:= Coefficients( b, x2*x1 );
x2:= x2/c1[1];
name:= "L2_2( ";
Append( name, LieAlgDBField2String( F ) );
Append( name, " )" );
K:= SolvableLieAlgebra( F, [2,2] );
f:= AlgebraHomomorphismByImages( K, L, Basis(K), [x1,x2] );
return rec( name:= [ name, [] ], isom:= f );
elif n = 3 then
x1:= b[1]; x2:= b[2];
if x1*x2 <> Zero(L) then
# Here [L,L] has dimension 1, and its centralizer
# has dimension 2, and it contains [L,L].
sp:= LieCentralizer( L, LieDerivedSubalgebra(L) );
b:= Basis( sp );
x1:= Basis(sp)[1];
x2:= Basis(sp)[2];
fi;
for k in [1..3] do
if not BL[k] in sp then
x3:= BL[k];
break;
fi;
od;
if x1*x3 = Zero(L) and x2*x3 = Zero(L) then
# Abelian!
name:= "L3_1( ";
Append( name, LieAlgDBField2String( F ) );
Append( name, " )" );
K:= SolvableLieAlgebra( F, [3,1] );
f:= AlgebraHomomorphismByImages( K, L, Basis(K), Basis(L) );
return rec( name:= [ name, [] ], isom:= f );
fi;
c1:= Coefficients( b, x3*x1 );
c2:= Coefficients( b, x3*x2 );
if c1[1] = c2[2] and c1[2] = Zero(F) and c2[1] = Zero(F) then
x3:= x3/c1[1];
name:= "L3_2( ";
Append( name, LieAlgDBField2String( F ) );
Append( name, " )" );
K:= SolvableLieAlgebra( F, [3,2] );
f:= AlgebraHomomorphismByImages( K, L, Basis(K), [x1,x2,x3] );
return rec( name:= [ name, [] ], isom:= f );
fi;
if c1[2] = Zero(F) then
# i.e., x1 is an eigenvector of ad x3, not good.
if c2[1] = Zero(F) then
# x2 is an eigenvector as well, so x1+x2 cannot be
# (by previous case)...
x1:= x1+x2;
else
x1:= x2;
fi;
fi;
x2:= x3*x1;
b:= Basis( Subspace( L, [x1,x2] ), [x1,x2] );
c2:= Coefficients( b, x3*x2 );
if c2[2] <> Zero(F) then
x2:= x2/c2[2];
x3:= x3/c2[2];
par:= c2[1]/(c2[2]^2);
name:= "L3_3( ";
Append( name, LieAlgDBField2String( F ) );
Append( name, ", " );
Append( name, String( par ) );
Append( name, " )" );
K:= SolvableLieAlgebra( F, [3,3,par] );
f:= AlgebraHomomorphismByImages( K, L, Basis(K), [x1,x2,x3] );
return rec( name:= [ name, [par] ], isom:= f );
fi;
par:= c2[1];
if HasIsFinite(F) and IsFinite(F) and not par in [Zero(F),One(F)] then
if Characteristic(F) = 2 then
f:= Inverse( FrobeniusAutomorphism( F ) );
a:= Image( f, par );
x2:= a*x2;
x3:= a*x3;
par:= a^2*par;
else
exp:= LogFFE( 1/par, PrimitiveRoot(F) );
if IsEvenInt( exp ) then
a:= PrimitiveRoot(F)^(exp/2);
x2:= a*x2;
x3:= a*x3;
par:= a^2*par;
else
exp:= LogFFE( PrimitiveRoot(F)/par, PrimitiveRoot(F) );
a:= PrimitiveRoot(F)^(exp/2);
x2:= a*x2;
x3:= a*x3;
par:= a^2*par;
fi;
fi;
fi;
name:= "L3_4( ";
Append( name, LieAlgDBField2String( F ) );
Append( name, ", " );
Append( name, String( par ) );
Append( name, " )" );
K:= SolvableLieAlgebra( F, [3,4,par] );
f:= AlgebraHomomorphismByImages( K, L, Basis(K), [x1,x2,x3] );
return rec( name:= [ name, [par] ], isom:= f );
elif n = 4 then
x1:= b[1]; x2:= b[2]; x3:= b[3];
for k in [1..4] do
if not BL[k] in sp then
x4:= BL[k];
break;
fi;
od;
K:= Subalgebra( L, [x1,x2,x3] );
adM:= List( Basis(K), x -> AdjointMatrix(Basis(K),x ) );
adK:= MutableBasis( F, [ 0*adM[1] ] );
bK:= [ ];
b_adK:= [ ];
for i in [1..Length(adM)] do
if not IsContainedInSpan( adK, adM[i] ) then
CloseMutableBasis( adK, adM[i] );
Add( bK, Basis(K)[i] );
Add( b_adK, adM[i] );
fi;
od;
adK:= VectorSpace( F, b_adK, 0*adM[1] );
mat:= List( Basis(K), x -> Coefficients( Basis(K), x4*x ) );
mat:= TransposedMat( mat );
if mat in adK then
c1:= Coefficients( Basis( adK, b_adK ), mat );
if c1 = [ ] then # L is abelian
name:= "L4_1( ";
Append( name, LieAlgDBField2String( F ) );
Append( name, " )" );
K:= SolvableLieAlgebra( F, [4,1] );
f:= AlgebraHomomorphismByImages( K, L, Basis(K), Basis(L) );
return rec( name:= [ name, [] ], isom:= f );
else
x3:= x4 - LinearCombination( c1, bK );
id:= LieAlgebraIdentification( K );
x4:= Image( id.isomorphism, Basis( Source( id.isomorphism ) )[3] );
x1:= Image( id.isomorphism, Basis( Source( id.isomorphism ) )[1] );
x2:= Image( id.isomorphism, Basis( Source( id.isomorphism ) )[2] );
K:= Subalgebra( L, [x1,x2,x3] );
fi;
fi;
id:= LieAlgebraIdentification( K );
x1:= Image( id.isomorphism, Basis( Source( id.isomorphism ) )[1] );
x2:= Image( id.isomorphism, Basis( Source( id.isomorphism ) )[2] );
x3:= Image( id.isomorphism, Basis( Source( id.isomorphism ) )[3] );
b:= Basis( K, [x1,x2,x3] );
c1:= Coefficients( b, x4*x1 );
c2:= Coefficients( b, x4*x2 );
c3:= Coefficients( b, x4*x3 );
mat:= [ c1, c2, c3 ];
num:= id.name[4];
if num = '1' then
# K is abelian
R:= PolynomialRing( F, 1 );
mat:= TransposedMat( mat );
mp:= MinimalPolynomial( F, mat );
is_diag:= true;
for i in [1..3] do
if not is_diag then break; fi;
for j in [1..3] do
if not is_diag then break; fi;
if i <> j and mat[i][j] <> Zero(F) then
is_diag:= false;
fi;
od;
od;
if Degree(mp) = 1 then
if c1[1] = Zero(F) then
name:= "L4_1( ";
Append( name, LieAlgDBField2String( F ) );
Append( name, " )" );
K:= SolvableLieAlgebra( F, [4,1] );
f:= AlgebraHomomorphismByImages( K, L, Basis(K), Basis(L) );
return rec( name:= [ name, [] ], isom:= f );
else
x4:= x4/c1[1];
name:= "L4_2( ";
Append( name, LieAlgDBField2String( F ) );
Append( name, " )" );
K:= SolvableLieAlgebra( F, [4,2] );
f:= AlgebraHomomorphismByImages(K,L,Basis(K),[x1,x2,x3,x4]);
return rec( name:= [ name, [] ], isom:= f );
fi;
fi;
if Degree(mp) = 2 then
# compute rational canonical form; rather clumsy algorithm
# which however works for the 3x3 case.
facs:= Factors( R, mp );
ev:= [ ];
for f in facs do
ef:= ExtRepPolynomialRatFun( f );
if Length( ef[1] ) = 0 then
Add( ev, -ef[2] );
else
Add( ev, Zero( F ) );
fi;
od;
if ev[1] <> ev[2] then
sol:= NullspaceMat( TransposedMat(mat) -
ev[1]*IdentityMat( 3, F ) );
ev1:= List( sol, x -> LinearCombination( x, [x1,x2,x3] ) );
sol:= NullspaceMat( TransposedMat(mat) -
ev[2]*IdentityMat( 3, F ) );
ev2:= List( sol, x -> LinearCombination( x, [x1,x2,x3] ) );
if Length( ev1 ) = 2 then
x1:= ev1[1];
x2:= ev1[2] + ev2[1];
x3:= x4*x2;
s:= ev[1]; t:= ev[2];
else
x1:= ev2[1];
x2:= ev2[2] + ev1[1];
x3:= x4*x2;
s:= ev[2]; t:= ev[1];
fi;
else
s:= ev[1]; t:= s;
sol:= NullspaceMat( TransposedMat(mat) -
ev[1]*IdentityMat( 3, F ) );
ev1:= List( sol, x -> LinearCombination( x, [x1,x2,x3] ) );
sp:= Subspace( L, ev1 );
if not x1 in sp then
x2:= x1;
elif not x3 in sp then
x2:= x3;
fi;
x3:= x4*x2;
sp:= Subspace( L, [x2,x3] );
if not ev1[1] in sp then
x1:= ev1[1];
else
x1:= ev1[2];
fi;
fi;
if s <> Zero(F) then
x3:= x3/s;
x4:= x4/s;
name:= "L4_3( ";
Append( name, LieAlgDBField2String( F ) );
Append( name, ", " );
Append( name, String( t/s ) );
Append( name, " )" );
K:= SolvableLieAlgebra( F, [4,3,t/s] );
f:= AlgebraHomomorphismByImages(K,L,Basis(K),
[x1,x2,x3,x4]);
return rec( name:= [ name, [t/s] ], isom:= f );
elif t <> Zero(F) then
x3:= x3/t;
x4:= x4/t;
name:= "L4_4( ";
Append( name, LieAlgDBField2String( F ) );
Append( name, " )" );
K:= SolvableLieAlgebra( F, [4,4] );
f:= AlgebraHomomorphismByImages(K,L,Basis(K),
[x1,x2,x3,x4]);
return rec( name:= [ name, [] ], isom:= f );
else
name:= "L4_5( ";
Append( name, LieAlgDBField2String( F ) );
Append( name, " )" );
K:= SolvableLieAlgebra( F, [4,5] );
f:= AlgebraHomomorphismByImages(K,L,Basis(K),
[x1,x2,x3,x4]);
return rec( name:= [ name, [] ], isom:= f );
fi;
fi;
# look for a cyclic vector...
# we take a random element, maybe better algorithm needed...
if IsFinite(F) then
cfs:= List( [1..10], x -> PrimitiveRoot(F)^x );
else
cfs:= List( [1..10], x -> x*One(F) );
fi;
Add( cfs, Zero(F) );
found:= false;
while not found do
u:= Random(cfs)*x1+Random(cfs)*x2+Random(cfs)*x3;
v:= x4*u;
w:= x4*v;
sp:= Subspace( L, [u,v,w] );
if Dimension(sp) = 3 then found:= true; fi;
od;
x1:= u; x2:= v; x3:= w;
b:= Basis( sp, [x1,x2,x3] );
c1:= Coefficients( b, x4*x3 );
if c1[3] <> Zero(F) then
x2:= x2/c1[3];
x3:= x3/(c1[3]^2);
x4:= x4/c1[3];
name:= "L4_6( ";
Append( name, LieAlgDBField2String( F ) );
Append( name, ", " );
Append( name, String( c1[1]/(c1[3]^3) ) );
Append( name, ", ");
Append( name, String( c1[2]/(c1[3]^2) ) );
Append( name, " )" );
K:= SolvableLieAlgebra( F, [4,6, c1[1]/(c1[3]^3), c1[2]/(c1[3]^2) ] );
f:= AlgebraHomomorphismByImages(K,L,Basis(K),[x1,x2,x3,x4]);
return rec( name:= [ name, [c1[1]/(c1[3]^3), c1[2]/(c1[3]^2)] ],
isom:= f );
else
if c1[1] <> Zero(F) and c1[2] <> Zero(F) then
s:= c1[2]/c1[1];
x2:= x2*s;
x3:= x3*s^2;
x4:= x4*s;
t:= s^2*c1[2];
name:= "L4_7( ";
Append( name, LieAlgDBField2String( F ) );
Append( name, ", " );
Append( name, String( t ) );
Append( name, ", ");
Append( name, String( t ) );
Append( name, " )" );
K:= SolvableLieAlgebra( F, [4,7, t, t] );
f:= AlgebraHomomorphismByImages(K,L,Basis(K),[x1,x2,x3,x4]);
return rec( name:= [ name, [t,t] ], isom:= f );
else
if HasIsFinite(F) and IsFinite(F) then
q:= Size(F);
if c1[1] <> Zero(F) then
par:= c1[1];
a:= PrimitiveRoot(F);
if q mod 6 = 1 or q mod 6 = 4 then
exp:= LogFFE( par, a );
b:= One(F);
if exp mod 3 <> 0 then
exp:= LogFFE( par/a, a );
b:= a;
if exp mod 3 <> 0 then
exp:= LogFFE( par/(a^2), a );
b:= a^2;
fi;
fi;
a:= a^(exp/3);
else
exp:= LogFFE( par, a );
if exp mod 3 = 1 then
for b in F do
if b^3 = a then
a:= a^((exp-1)/3)*b;
break;
fi;
od;
elif exp mod 3 = 2 then
for b in F do
if b^3 = a^2 then
a:= a^((exp-2)/3)*b;
break;
fi;
od;
else
a:= a^(exp/3);
fi;
b:= One(F);
fi;
c1:= [ b, Zero(F) ];
elif c1[2] <> Zero(F) then
par:= c1[2];
a:= PrimitiveRoot(F);
if IsEvenInt( q ) then
exp:= LogFFE( par, a );
if exp mod 2 = 1 then
a:= a^((exp-1)/2)*a^(q/2);
else
a:= a^(exp/2);
fi;
b:= One(F);
else
exp:= LogFFE( par, a );
b:= One(F);
if not IsEvenInt( exp ) then
exp:= LogFFE( par/a, a );
b:= a;
fi;
a:= a^(exp/2);
fi;
c1:= [ Zero(F), b ];
else
a:= One(F);
fi;
x2:= x2/a;
x3:= x3/(a^2);
x4:= x4/a;
fi;
name:= "L4_7( ";
Append( name, LieAlgDBField2String( F ) );
Append( name, ", " );
Append( name, String( c1[1] ) );
Append( name, ", ");
Append( name, String( c1[2] ) );
Append( name, " )" );
K:= SolvableLieAlgebra( F, [4,7, c1[1], c1[2] ] );
f:= AlgebraHomomorphismByImages(K,L,Basis(K),[x1,x2,x3,x4]);
return rec( name:= [ name, [c1[1],c1[2]] ], isom:= f );
fi;
fi;
elif num = '2' then
D:= TransposedMat( mat );
x4:= x4+D[1][3]*x1+D[2][3]*x2-D[2][2]*x3;
mat:= List( [x1,x2,x3], x -> Coefficients( b, x4*x ) );
D:= TransposedMat( mat );
if D[1][1] <> Zero(F) then
x4:= x4/D[1][1];
D:= D/D[1][1];
w:= D[1][2];
v:= D[2][1];
if w <> Zero(F) then
x1:= w*x1;
v:= v*w;
return LieAlgDBHelper.isomM9( L, x1, x2, x3, x4, v );
else
#direct sum...
dd:= DirectSumDecomposition( L );
id:= LieAlgebraIdentification( dd[1] );
f:= id.isomorphism;
x1:= Image( f, Basis( Source(f) )[2] );
x2:= Image( f, Basis( Source(f) )[1] );
id:= LieAlgebraIdentification( dd[2] );
f:= id.isomorphismorphism;
x3:= Image( f, Basis( Source(f) )[2] );
x4:= Image( f, Basis( Source(f) )[1] );
name:= "L4_8( ";
Append( name, LieAlgDBField2String( F ) );
Append( name, " )" );
K:= SolvableLieAlgebra( F, [4,8] );
f:= AlgebraHomomorphismByImages(K,L,Basis(K),
[x1,x2,x3,x4]);
return rec( name:= [ name, [] ], isom:= f );
fi;
else
if D[2][1] <> Zero(F) then
x4:= x4/D[2][1];
a:= D[1][2]/D[2][1];
if Characteristic(F) <> 2 then
a:= a - (1/4)*One(F);
return LieAlgDBHelper.isomM9( L, x1/(4*One(F))+x2/(2*One(F)),
x1/(2*One(F)), x3, x3/(2*One(F))+x4, a );
else
return LieAlgDBHelper.isomM10( L, x1, x2, x3, x4, a );
fi;
else
if D[1][2] <> Zero(F) then
x4:= x4/D[1][2];
if Characteristic(F) <> 2 then
y1:= (x1+x2)/(2*One(F));
y2:= x2;
y3:= x3;
y4:= (x3+x4)/(2*One(F));
return LieAlgDBHelper.isomN( L, y1, y2, y3, y4 );
else
name:= "L4_10( ";
Append( name, LieAlgDBField2String( F ) );
Append( name, ", " );
Append( name, String( Zero(F) ) );
Append( name, " )" );
K:= SolvableLieAlgebra( F, [4,10, Zero(F) ] );
f:= AlgebraHomomorphismByImages(K,L,Basis(K),
[x2,x1,x3,x4]);
return rec( name:= [ name, [Zero(F)] ], isom:= f );
fi;
fi;
fi;
fi;
elif num = '3' then
D:= TransposedMat( mat );
a:= id.parameters[1];
if a <> Zero(F) then
x4:= x4-(D[1][3]/a-D[2][3])*x1+D[1][3]/a*x2-D[2][1]*x3;
x4:= x4/D[1][1];
return LieAlgDBHelper.isomM9( L, x2, x1, x4, x3, a );
else
x4:= x4+D[2][3]*x1-D[2][1]*x3;
mat:= List( [x1,x2,x3], x -> Coefficients( b, x4*x ) );
D:= TransposedMat( mat );
if D[1][3] = Zero(F) then
x4:= x4/D[1][1];
return LieAlgDBHelper.isomM9( L, x2, x1, x4, x3, a );
elif D[1][1] = Zero(F) then
x4:= x4/D[1][3];
name:= "L4_6( ";
Append( name, LieAlgDBField2String( F ) );
Append( name, ", " );
Append( name, String( Zero(F) ) );
Append( name, ", ");
Append( name, String( Zero(F) ) );
Append( name, " )" );
K:= SolvableLieAlgebra( F, [4,6, Zero(F), Zero(F)] );
f:= AlgebraHomomorphismByImages(K,L,Basis(K),
[-x4,x1,x2,x3]);
return rec( name:= [ name, [Zero(F),Zero(F)] ], isom:= f );
else
x4:= x4/D[1][1];
x3:= -D[1][3]/D[1][1]*x1+x3;
return LieAlgDBHelper.isomM9( L, x2, x1, x4, x3, a );
fi;
fi;
else # i.e., num = '4'...
D:= TransposedMat( mat );
a:= id.parameters[1];
if a <> Zero(F) and Characteristic(F) <> 2 then
x4:= x4+D[2][3]*x1+D[1][3]/a*x2-D[2][1]*x3;
x4:= x4/D[1][1];
return LieAlgDBHelper.isomM9( L, x1/(4*One(F))+x2/(2*One(F)), x1/(2*One(F)),
x4, x3+x4/(2*One(F)), a-(1/4*One(F)) );
elif a <> Zero(F) and Characteristic(F) = 2 then
x4:= x4+D[2][3]*x1+D[1][3]/a*x2-D[2][1]*x3;
if D[3][3] = Zero(F) then
x4:= x4/D[1][1];
return LieAlgDBHelper.isomM10( L, x1, x2, x4, x3, a );
elif D[1][1] <> Zero(F) then
x4:= x4/D[1][1];
b:= One(F) + D[3][3]/D[1][1];
return LieAlgDBHelper.isomM11( L, x1, x2, x3, x4, a, b );
else
if a <> One(F) then
x4:= x4/D[3][3];
y1:= (a*x1+x2)/(a+One(F));
y2:= a*(x1+x2)/(a+One(F));
y3:= x3;
y4:= x3+(a+1)*x4;
return LieAlgDBHelper.isomM11( L, y1, y2, y3, y4, a, a );
else
return LieAlgDBHelper.isomM11( L, x2, x1, x3, x4, One(F), Zero(F) );
fi;
fi;
else
x4:= x4+D[2][3]*x1-D[2][1]*x3;
if D[1][3]=Zero(F) and D[3][1]=Zero(F) and D[1][1]=D[3][3] then
x4:= x4/D[1][1];
name:= "L4_12( ";
Append( name, LieAlgDBField2String( F ) );
Append( name, " )" );
K:= SolvableLieAlgebra( F, [4,12] );
f:= AlgebraHomomorphismByImages(K,L,Basis(K),
[x1,x2,x3,x4]);
return rec( name:= [ name, [] ], isom:= f );
else
if D[3][3] <> Zero(F) then
u1:= D[1][1];
u3:= D[3][1];
v1:= D[1][3];
v3:= D[3][3];
if v1 <> Zero(F) then
x1:= x1+(v3/v1)*x3;
elif u3 <> Zero(F) then
x3:= -(v3/u3)*x1+x3;
elif u1 = Zero(F) then
y1:= x1;
x1:= x3;
x2:= -x2;
x3:= y1;
else
y1:= x1;
x1:= x1/(One(F) - v3/u1)-x3;
x3:= -(v3/u1)*y1/(One(F) - v3/u1)+x3;
fi;
K:= Subalgebra( L, [x1,x2,x3] );
b:= Basis( K, [x1,x2,x3] );
c1:= Coefficients( b, x4*x1 );
c2:= Coefficients( b, x4*x2 );
c3:= Coefficients( b, x4*x3 );
mat:= [ c1, c2, c3 ];
D:= TransposedMat( mat );
fi; # now D[3][3] is zero...
u1:= D[1][1]; u3:= D[3][1]; v1:= D[1][3];
if u1 <> Zero(F) and v1 <> Zero(F) then
x4:= x4/u1;
x1:= (1/u1)*x1;
x2:= (1/(u1*v1))*x2;
x3:= (1/v1)*x3;
a:= (v1/u1)*(u3/u1);
name:= "L4_13( ";
Append( name, LieAlgDBField2String( F ) );
Append( name, ", " );
Append( name, String( a ) );
Append( name, " )" );
K:= SolvableLieAlgebra( F, [4,13, a] );
f:= AlgebraHomomorphismByImages(K,L,Basis(K),
[x1,x2,x3,x4]);
return rec( name:= [ name, [a] ], isom:= f );
elif u1 <> Zero(F) and v1 = Zero(F) then
if u3 <> Zero(F) then
x4:= x4/u1;
x1:= (1/u3)*x1;
x2:= (1/(u1*u3))*x2;
x3:= (1/u1)*x3;
y1:= x1;
x1:= x1+x3;
x2:= -x2;
x3:= y1;
else
x4:= x4/u1;
x3:= x1+x3;
fi;
name:= "L4_13( ";
Append( name, LieAlgDBField2String( F ) );
Append( name, ", " );
Append( name, String( Zero(F) ) );
Append( name, " )" );
K:= SolvableLieAlgebra( F, [4,13, Zero(F)] );
f:= AlgebraHomomorphismByImages(K,L,Basis(K),
[x1,x2,x3,x4]);
return rec( name:= [ name, [Zero(F)] ], isom:= f );
elif u1 = Zero(F) and v1 <> Zero(F) then
x4:= x4/v1;
par:= u3/v1;
if HasIsFinite(F) and IsFinite(F) and
not par in [Zero(F),One(F)] then
if Characteristic(F) = 2 then
f:= Inverse( FrobeniusAutomorphism( F ) );
a:= Image( f, 1/par );
else
exp:= LogFFE( 1/par, PrimitiveRoot(F) );
if IsEvenInt( exp ) then
a:= PrimitiveRoot(F)^(exp/2);
else
exp:= LogFFE( PrimitiveRoot(F)/par,
PrimitiveRoot(F) );
a:= PrimitiveRoot(F)^(exp/2);
fi;
fi;
x1:= a*x1;
x2:= a*x2;
x4:= a*x4;
par:= a^2*par;
fi;
if par = Zero(F) then
name:= "L4_7( ";
Append( name, LieAlgDBField2String( F ) );
Append( name, ", " );
Append( name, String( Zero(F) ) );
Append( name, ", " );
Append( name, String( Zero(F) ) );
Append( name, " )" );
K:= SolvableLieAlgebra( F, [4,7, Zero(F), Zero(F)] );
f:= AlgebraHomomorphismByImages( K, L, Basis(K),
[-x4,x1,x2,x3]);
return rec( name:= [ name, [Zero(F),Zero(F)] ],
isom:= f );
else
name:= "L4_14( ";
Append( name, LieAlgDBField2String( F ) );
Append( name, ", " );
Append( name, String( par ) );
Append( name, " )" );
K:= SolvableLieAlgebra( F, [4,14, par] );
f:= AlgebraHomomorphismByImages( K, L, Basis(K),
[x1,x2,x3,x4] );
return rec( name:= [ name, [par] ], isom:= f );
fi;
else
x4:= x4/u3;
y1:= x3;
y2:= -x2;
y3:= x1;
y4:= x4;
name:= "L4_7( ";
Append( name, LieAlgDBField2String( F ) );
Append( name, ", " );
Append( name, String( Zero(F) ) );
Append( name, ", " );
Append( name, String( Zero(F) ) );
Append( name, " )" );
K:= SolvableLieAlgebra( F, [4, 7, Zero(F), Zero(F)] );
f:= AlgebraHomomorphismByImages( K, L, Basis(K),
[-y4,y1,y2,y3] );
return rec( name:= [ name, [Zero(F),Zero(F)] ], isom:= f );
fi;
fi;
fi;
fi;
else
Error( "dim > 4 not yet implemented" );
fi;
end;
#========================================================================
LieAlgDBHelper.liealg_hom:= function( K, L, p, i )
return AlgebraHomomorphismByImagesNC( K, L, p, i );
end;
LieAlgDBHelper.class_dim_le4:= function( L )
# finds an isomorphism of the nilpotent Lie algebra L of dim <= 4,
# to a "normal form".
local F, C, C1, D, i, x, y, z, u, T, V, K, A, cc, b, a;
F:= LeftActingDomain( L );
if Dimension( L ) = 3 then
if IsLieAbelian( L ) then
return rec( type:= [ 3, 1 ], f:= LieAlgDBHelper.liealg_hom( L, L, Basis( L ), Basis( L ) ) );
else
C:= LieCentre( L );
for i in [1..3] do
if not Basis( L )[i] in C then
x:= Basis( L )[i]; break;
fi;
od;
C:= Subalgebra( L, [x,Basis( C )[1]] );
for i in [1..3] do
if not Basis( L )[i] in C then
y:= Basis( L )[i]; break;
fi;
od;
T:= EmptySCTable( 3, Zero( F ), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
K:= LieAlgebraByStructureConstants( F, T );
return rec( type:= [ 3, 2 ], f:= LieAlgDBHelper.liealg_hom( L, K, [x,y,x*y], Basis( K ) ) );
fi;
elif Dimension( L ) = 4 then
if IsLieAbelian( L ) then
return rec( type:= [ 4, 1 ], f:= LieAlgDBHelper.liealg_hom( L, L, Basis( L ), Basis( L ) ) );
else
C:= LieCentre( L );
if Dimension( C ) = 2 then
for i in [1..4] do
if not Basis( L )[i] in C then
x:= Basis( L )[i]; break;
fi;
od;
C1:= Subalgebra( L, [ Basis( C )[1], Basis( C )[2], x ] );
for i in [1..4] do
if not Basis( L )[i] in C1 then
y:= Basis( L )[i]; break;
fi;
od;
A:= Subalgebra( L, [ x*y ] );
if Basis( C )[1] in A then
u:= Basis( C )[2];
else
u:= Basis( C )[1];
fi;
T:= EmptySCTable( 4, Zero( F ), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
K:= LieAlgebraByStructureConstants( F, T );
return rec( type:= [ 4, 2 ], f:= LieAlgDBHelper.liealg_hom( L, K, [x,y,x*y,u], Basis( K ) ) );
elif Dimension( C ) = 1 then
D:= LieDerivedSubalgebra( L );
b:= ShallowCopy( Basis( D ) );
cc:= [ ];
for i in [1..4] do
x:= Basis( L )[i];
if not x in D then
Add( cc, x );
Add( b, x );
D:= Subalgebra( L, b );
fi;
od;
z:= cc[1]*cc[2];
if cc[1]*z <> Zero( L ) then
x:= cc[1]; y:= cc[2];
else
x:= cc[2]; y:= cc[1];
fi;
z:= x*y; u:= x*z;
# we have to change y to make sure that [x,y]= z, and
# [y,u]= 0.
V:= Basis( Subalgebra( L, [u] ), [u] );
a:= Coefficients( V, y*z )[1];
y:= -a*x+y;
T:= EmptySCTable( 4, Zero( F ), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,4] );
K:= LieAlgebraByStructureConstants( F, T );
return rec( type:= [ 4, 3 ], f:= LieAlgDBHelper.liealg_hom( L, K, [x,y,z,u], Basis( K ) ) );
fi;
fi;
fi;
end;
LieAlgDBHelper.skew_symm_NF:= function( V, f )
# here V is a vector space, and f : VxV -> F a skew-symmetric
# bilinear function; we compute a basis of V such that f has standard form.
# Here f is just given as a GAP function.
local b, dim, found, i, j, c, W, bU, u, v, bb, done, x, w;
b:= Basis(V);
dim:= Dimension(V);
found:= false;
for i in [1..dim] do
if found then break; fi;
for j in [i+1..dim] do
c:= f(b[i],b[j]);
if not IsZero(c) then
u:= [b[i]]; v:= [b[j]/c];
found:= true; break;
fi;
od;
od;
done:= false;
while not done do
# consider the linear map T: V --> V defined by
# T(w) = \sum_i f(w,v[i])u[i] - f(w,u[i])v[i],
# then T is projection onto the space spanned by the elts in u,v.
# set U = (1-T)V; then T is zero on U, and V is the direct
# sum of U and the space spanned by u,v. We compute U.
W:= MutableBasis( LeftActingDomain(V), [], Zero(V) );
for x in b do
w:= x-Sum( List( [1..Length(u)], k -> f(x,v[k])*u[k]-f(x,u[k])*v[k] ) );
if not IsContainedInSpan( W, w ) then
CloseMutableBasis( W, w );
fi;
od;
bU:= BasisVectors(W);
found:= false;
for i in [1..Length(bU)] do
if found then break; fi;
for j in [i+1..Length(bU)] do
c:= f(bU[i],bU[j]);
if not IsZero(c) then
Add( u, bU[i] ); Add( v, bU[j]/c );
found:= true; break;
fi;
od;
od;
if not found then
done:= true;
bb:= [ ];
for i in [1..Length(u)] do
Add( bb, u[i] );
Add( bb, v[i] );
od;
Append( bb, bU );
elif 2*Length(u) = dim then
done:= true;
bb:= [ ];
for i in [1..Length(u)] do
Add( bb, u[i] );
Add( bb, v[i] );
od;
fi;
od;
return bb;
end;
LieAlgDBHelper.class_dim_5:= function( K )
local F, C, D, ind, i, bL, bsp, L, W, t, T, d, N, imgs, p, s, coc1, coc2, tau, M, coc21, coc22,
bM, cz, cx, type, a, b, c, m, mat, cf, c1, c2, x, mt, sp, y, y1, r;
if IsLieAbelian( K ) then
return rec( type:= [ 5, 1 ], f:= LieAlgDBHelper.liealg_hom( K, K, Basis( K ), Basis( K ) ) );
fi;
F:= LeftActingDomain( K );
# see if there is an abelian component ( necessarily of dim 1 )
C:= LieCentre( K );
D:= LieDerivedSubalgebra( K );
ind:= 0;
for i in [1..Dimension( C )] do
if not Basis( C )[i] in D then
ind:= i;
break;
fi;
od;
if ind > 0 then
bL:= ShallowCopy( Basis( D ) );
bsp:= ShallowCopy( bL );
Add( bsp, Basis( C )[ind] );
sp:= MutableBasis( F, bsp );
for i in [1..Dimension( K )] do
if not IsContainedInSpan( sp, Basis( K )[i] ) then
Add( bL, Basis( K )[i] );
CloseMutableBasis( sp, Basis( K )[i] );
fi;
od;
# now K = bL \oplus C[ind].
L:= Subalgebra( K, bL );
bL:= ShallowCopy( Basis( L ) );
Add( bL, Basis( C )[ind] );
W:= Basis( K, bL );
# so now bL is a basis of K, the first n-1 elements form
# a basis of an ideal, the last element of an abelian ideal.
# We have an isomorphism of K to the direct sum of L with a
# 1-dim ideal; write x\in K on the basis bL.
r:= LieAlgDBHelper.class_dim_le4( L ); type:= r.type; tau:= r.f;
T:= ShallowCopy( StructureConstantsTable( Basis( Range( tau ) ) ) );
d:= Dimension( Range( tau ) );
for i in [1..d] do
T[i]:= ShallowCopy( T[i] );
Add( T[i], [ [], [] ] );
od;
T[d+3]:= T[d+2]; T[d+2]:= T[d+1];
T[d+1]:= List( [1..d+1], x -> [ [], [] ] );
N:= LieAlgebraByStructureConstants( F, T );
imgs:= [ ];
for x in Basis( K ) do
cx:= Coefficients( W, x );
y:= Image( tau, LinearCombination( Basis( L ), cx{[1..Length( cx )-1]} ) );
y:= ShallowCopy( Coefficients( Basis( Range( tau ) ), y ) );
Add( y, cx[ Length( cx ) ] );
if type[2]= 2 then
# isom
y1:= ShallowCopy( y );
y1[1]:= y[1];
y1[2]:= y[2];
y1[3]:= y[3];
y1[4]:= y[5];
y1[5]:= y[4];
y:= ShallowCopy( y1 );
fi;
Add( imgs, LinearCombination( Basis( N ), y ) );
od;
return rec( type:= [ 5, type[2] ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) );
fi;
# if there are no central components, then we look at the
# cocycle we get by dividing by the centre, and so on.
C:= LieCentre( K );
p:= NaturalHomomorphismByIdeal( K, C );
L:= Range( p );
s:= function( v ) return PreImagesRepresentative( p, v ); end;
coc1:= function( u, v ) return s( u )*s( v )-s( u*v ); end;
r:= LieAlgDBHelper.class_dim_le4( L ); type:= r.type; tau:= r.f;
M:= Range( tau );
coc2:= function( u, v ) return coc1( PreImage( tau,u ), PreImage( tau,v ) ); end;
# i.e., the cocycle for M
# now we need to map coc2 to normal form, according to the type of M
if type = [ 4, 1 ] then
coc21:= function( u, v ) return Coefficients( Basis( C ), coc2( u,v ) )[1]; end;
bM:= Basis( M, LieAlgDBHelper.skew_symm_NF( M, coc21 ) );
T:= EmptySCTable( 5, Zero( F ), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,5] );
SetEntrySCTable( T, 3, 4, [1,5] );
N:= LieAlgebraByStructureConstants( F, T );
imgs:= [ ];
for x in Basis( K ) do
# the coordinate of the central part of x:
# this will be the coordinate of the central part of the
# image of x.
cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) )[1];
cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p,x ) ) ) );
Add( cx, cz );
Add( imgs, LinearCombination( Basis( N ), cx ) );
od;
return rec( type:= [ 5, 4 ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) );
elif type = [ 4, 2 ] then
coc21:= function( u, v ) return Coefficients( Basis( C ), coc2( u,v ) )[1]; end;
bM:= ShallowCopy( BasisVectors( Basis( M ) ) );
a:= coc21( bM[1], bM[3] );
b:= coc21( bM[1], bM[4] );
c:= coc21( bM[2], bM[3] );
d:= coc21( bM[2], bM[4] );
mat:= IdentityMat( 4, F );
if IsZero( a ) then
# c is non zero
m:= IdentityMat( 4, F );
m[2][1]:= 1;
mat:= m;
a:= c; b:= b+d;
fi;
m:= IdentityMat( 4, F );
m[1][1]:= 1/a;
m[1][2]:= -c;
m[2][2]:= a;
m[4][4]:= 1/( a*d-b*c );
m[3][4]:= -m[4][4]*b/a;
mat:= mat*m;
mat:= TransposedMat( mat );
bM:= Basis( M, List( mat, x -> LinearCombination( Basis( M ), x ) ) );
T:= EmptySCTable( 5, Zero( F ), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,5] );
SetEntrySCTable( T, 2, 4, [1,5] );
N:= LieAlgebraByStructureConstants( F, T );
imgs:= [ ];
for x in Basis( K ) do
cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) )[1];
cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) );
Add( cx, cz );
# effectively subtract a coboundary ( i.e., apply one further
# isomorphism ):
# Here we have ( in the normal form Lie algebra )
# [x1,x2]= x3. Hence c( x1,x2 )= 1, rest 0 is a coboundary.
# The isomorphism corresponding to the subtraction of c
# from the cocycle, subtracts coc21( x1, x2 )*x5 from
# the element. ( Rather vague comment... ).
cf:= coc21( bM[1], bM[2] );
cx[5]:= cx[5] - cf*cx[3];
Add( imgs, LinearCombination( Basis( N ), cx ) );
od;
return rec( type:= [ 5, 5 ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) );
elif type = [ 4, 3 ] then
coc21:= function( u, v ) return Coefficients( Basis( C ), coc2( u,v ) )[1]; end;
bM:= Basis( M );
a:= coc21( bM[1], bM[4] );
b:= coc21( bM[2], bM[3] );
if not IsZero( b ) then
T:= EmptySCTable( 5, Zero( F ), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,4] );
SetEntrySCTable( T, 1, 4, [1,5] );
SetEntrySCTable( T, 2, 3, [1,5] );
N:= LieAlgebraByStructureConstants( F, T );
imgs:= [ ];
for x in Basis( K ) do
cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) )[1];
cx:= ShallowCopy( Coefficients( Basis( M ), Image( tau, Image( p, x ) ) ) );
cf:= a/b;
cx[1]:= cx[1]*cf; cx[2]:= cx[2]*cf; cx[3]:= cx[3]*cf^2; cx[4]:= cx[4]*cf^3;
Add( cx, cf^4*cz/a );
# take care of the coboundary...
c1:= coc21( cf^-1*Basis( M )[1],cf^-1*Basis( M )[2] );
cx[5]:= cx[5] - c1*( cf^4/a )*cx[3];
c1:= coc21( cf^-1*Basis( M )[1],cf^-2*Basis( M )[3] );
cx[5]:= cx[5] - c1*( cf^4/a )*cx[4];
Add( imgs, LinearCombination( Basis( N ), cx ) );
od;
return rec( type:= [ 5, 6 ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) );
else
T:= EmptySCTable( 5, Zero( F ), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,4] );
SetEntrySCTable( T, 1, 4, [1,5] );
N:= LieAlgebraByStructureConstants( F, T );
imgs:= [ ];
for x in Basis( K ) do
cz:= Coefficients( Basis( C ), x-s( Image( p, x ) ) )[1];
cx:= ShallowCopy( Coefficients( Basis( M ), Image( tau, Image( p,x ) ) ) );
Add( cx, cz/a );
# take care of the coboundary...
cf:= coc21( Basis( M )[1], Basis( M )[2] );
cx[5]:= cx[5] - cf/a*cx[3];
cf:= coc21( Basis( M )[1], Basis( M )[3] );
cx[5]:= cx[5] - cf/a*cx[4];
Add( imgs, LinearCombination( Basis( N ), cx ) );
od;
return rec( type:= [ 5, 7 ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) );
fi;
elif type = [ 3, 1 ] then
coc21:= function( u, v ) return Coefficients( Basis( C ), coc2( u,v ) )[1]; end;
coc22:= function( u, v ) return Coefficients( Basis( C ), coc2( u,v ) )[2]; end;
bM:= LieAlgDBHelper.skew_symm_NF( M, coc21 );
b:= coc22( bM[1], bM[3] ); c:= coc22( bM[2],bM[3] );
if IsZero( b ) then
# change the basis bM so that it becomes nonzero.
bM[1]:= bM[1]+bM[2];
b:= coc22( bM[1],bM[3] ); c:= coc22( bM[2],bM[3] );
fi;
# change the basis so that c becomes zero...
bM[2]:= bM[2]-c/b*bM[1];
bM:= Basis( M, bM );
a:= coc22( bM[1],bM[2] );
b:= coc22( bM[1],bM[3] );
T:= EmptySCTable( 5, Zero( F ), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,4] );
SetEntrySCTable( T, 1, 3, [1,5] );
N:= LieAlgebraByStructureConstants( F, T );
imgs:= [ ];
for x in Basis( K ) do
# the coordinate of the central part of x:
cz:= Coefficients( Basis( C ), x-s( Image( p, x ) ) );
cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) );
# account for the base change in the centre:
cz[2]:= -cz[1]*a/b+cz[2]/b;
Append( cx, cz );
Add( imgs, LinearCombination( Basis( N ), cx ) );
od;
return rec( type:= [ 5, 8 ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) );
elif type = [ 3, 2 ] then
coc21:= function( u, v ) return Coefficients( Basis( C ), coc2( u,v ) )[1]; end;
coc22:= function( u, v ) return Coefficients( Basis( C ), coc2( u,v ) )[2]; end;
bM:= Basis( M );
m:= [[coc21( bM[1],bM[3] ),coc21( bM[2],bM[3] )],
[coc22( bM[1],bM[3] ),coc22( bM[2],bM[3] )] ];
m:= m^-1;
mt:= TransposedMat( m );
T:= EmptySCTable( 5, Zero( F ), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,4] );
SetEntrySCTable( T, 2, 3, [1,5] );
N:= LieAlgebraByStructureConstants( F, T );
c1:= mt[1][1]*coc21( bM[1],bM[2] )+mt[2][1]*coc22( bM[1],bM[2] );
c2:= mt[1][2]*coc21( bM[1],bM[2] )+mt[2][2]*coc22( bM[1],bM[2] );
imgs:= [ ];
for x in Basis( K ) do
cz:= Coefficients( Basis( C ), x-s( Image( p, x ) ) );
cx:= ShallowCopy( Coefficients( Basis( M ), Image( tau, Image( p, x ) ) ) );
Append( cx, cz*mt );
# take care of the coboundary...
cx[4]:= cx[4] - c1*cx[3];
cx[5]:= cx[5] - c2*cx[3];
Add( imgs, LinearCombination( Basis( N ), cx ) );
od;
return rec( type:= [ 5, 9 ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) );
fi;
end;
LieAlgDBHelper.comp_mat:= function( x, y, det )
local a,b,c,d;
if IsZero(y) then
c:= 0; a:= 1/x; d:= x*det;b:= 0;
else
a:= -y*det; c:= -y*det; d:= x*det; b:= (1+x*y*det)/y;
fi;
return [[a,b],[c,d]];
end;
LieAlgDBHelper.sq_free:= function( r )
# r is a rational number, we output a list of two rationals [a,b],
# such that b is square free, r = a^2b and b is an integer.
local sign, f, u, v, k, n1, n2, d1, d2;
if r >= 0 then
sign:= 1;
else
sign:= -1;
r:= -r;
fi;
f:= Factors( NumeratorRat( r ) );
u:= [ ]; v:= [ ];
k:= 1;
while k <= Length( f ) do
if k < Length( f ) then
if f[k]= f[k+1] then
Add( u, f[k] );
k:= k+2;
else
Add( v, f[k] );
k:= k+1;
fi;
else
Add( v, f[k] );
k:= k+1;
fi;
od;
n1:= Product( u ); n2:= Product( v );
f:= Factors( DenominatorRat( r ) );
u:= [ ]; v:= [ ];
k:= 1;
while k <= Length( f ) do
if k < Length( f ) then
if f[k]= f[k+1] then
Add( u, f[k] );
k:= k+2;
else
Add( v, f[k] );
k:= k+1;
fi;
else
Add( v, f[k] );
k:= k+1;
fi;
od;
d1:= Product( u ); d2:= Product( v );
return [n1/(d1*d2), sign*n2*d2 ];
end;
LieAlgDBHelper.class_dim_6:= function( K )
local F, C, D, ind, i, bL, bsp, sp, W, r, T, d, N, imgs, x, cx, g, P, p, L,
s, coc1, coc2, coc21, coc22, coc23, type, tau, M, bM, aa, a, bb, b, c, m,
v1, v2, e, f, c1, c2, c3, c4, a21, tt, cz1, cx1, s13, s24, s14,
t, uu, u, v, cf, y, y1, cz, a32, a41, a42, store, q, cc, f1, f2, f3, f4,
f5, ff, cf1, xi, ex, eps,root;
if IsLieAbelian( K ) then
return rec( type:= [ 6, 1 ], f:= LieAlgDBHelper.liealg_hom( K, K, Basis( K ), Basis( K ) ) );
fi;
F:= LeftActingDomain( K );
# see if there is an abelian component ( necessarily of dim 1 )
C:= LieCentre( K );
D:= LieDerivedSubalgebra( K );
ind:= 0;
for i in [1..Dimension( C )] do
if not Basis( C )[i] in D then
ind:= i;
break;
fi;
od;
if ind > 0 then
bL:= ShallowCopy( Basis( D ) );
bsp:= ShallowCopy( bL );
Add( bsp, Basis( C )[ind] );
sp:= MutableBasis( F, bsp );
for i in [1..Dimension( K )] do
if not IsContainedInSpan( sp, Basis( K )[i] ) then
Add( bL, Basis( K )[i] );
CloseMutableBasis( sp, Basis( K )[i] );
fi;
od;
# now K = bL \oplus C[ind].
L:= Subalgebra( K, bL );
bL:= ShallowCopy( Basis( L ) );
Add( bL, Basis( C )[ind] );
W:= Basis( K, bL );
# so now bL is a basis of K, the first n-1 elements form
# a basis of an ideal, the last element of an abelian ideal.
# We have an isomorphism of K to the direct sum of L with a
# 1-dim ideal; write x\in K on the basis bL.
r:= LieAlgDBHelper.class_dim_5( L ); type:= r.type; tau:= r.f;
T:= ShallowCopy( StructureConstantsTable( Basis( Range( tau ) ) ) );
d:= Dimension( Range( tau ) );
for i in [1..d] do
T[i]:= ShallowCopy( T[i] );
Add( T[i], [ [], [] ] );
od;
T[d+3]:= T[d+2]; T[d+2]:= T[d+1];
T[d+1]:= List( [1..d+1], x -> [ [], [] ] );
N:= LieAlgebraByStructureConstants( F, T );
imgs:= [ ];
for x in Basis( K ) do
cx:= Coefficients( W, x );
y:= Image( tau, LinearCombination( Basis( L ), cx{[1..Length( cx )-1]} ) );
y:= ShallowCopy( Coefficients( Basis( Range( tau ) ), y ) );
Add( y, cx[ Length( cx ) ] );
if type[2]= 2 then
# isom
y1:= ShallowCopy( y );
y1[1]:= y[1];
y1[2]:= y[2];
y1[3]:= y[3];
y1[4]:= y[6];
y1[5]:= y[4];
y1[6]:= y[5];
y:= ShallowCopy( y1 );
elif type[2]= 3 then
# isom
y1:= ShallowCopy( y );
y1[1]:= y[1];
y1[2]:= y[2];
y1[3]:= y[3];
y1[4]:= y[4];
y1[5]:= y[6];
y1[6]:= y[5];
y:= ShallowCopy( y1 );
fi;
Add( imgs, LinearCombination( Basis( N ), y ) );
od;
return rec( type:= [ 6, type[2] ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) );
fi;
# if there are no central components, then we look at the
# cocycle we get by dividing by the centre, and so on.
C:= LieCentre( K );
p:= NaturalHomomorphismByIdeal( K, C );
L:= Range( p );
s:= function( v ) return PreImagesRepresentative( p, v ); end;
coc1:= function( u, v ) return s( u )*s( v )-s( u*v ); end;
if Dimension( L ) <= 4 then
r:= LieAlgDBHelper.class_dim_le4( L );
else
r:= LieAlgDBHelper.class_dim_5( L );
fi;
type:= r.type; tau:= r.f;
M:= Range( tau );
coc2:= function( u, v ) return coc1( PreImage( tau,u ), PreImage( tau,v ) ); end;
# i.e., the cocycle for M
# now we need to map coc2 to normal form, according to the type of M
if type = [ 5, 2 ] then
coc21:= function( u, v ) return Coefficients( Basis( C ), coc2( u,v ) )[1]; end;
bM:= ShallowCopy( BasisVectors( Basis( ( M ) ) ) );
# we change the basis bM so that the cocycle is in "normal form":
a:= coc21( bM[1],bM[3] ); d:= coc21( bM[2],bM[3] );
m:= LieAlgDBHelper.comp_mat( a, d, One( F ) );
v1:= m[1][1]*bM[1]+m[1][2]*bM[2];
v2:= m[2][1]*bM[1]+m[2][2]*bM[2];
bM[1]:= v1; bM[2]:= v2;
b:= coc21( bM[1],bM[4] ); c:= coc21( bM[1],bM[5] );
bM[4]:= bM[4]-b*bM[3];
bM[5]:= bM[5]-c*bM[3];
e:= coc21( bM[2],bM[4] ); f:= coc21( bM[2],bM[5] );
g:= coc21( bM[4],bM[5] );
if not ( IsZero( e ) and IsZero( f ) ) then
m:= LieAlgDBHelper.comp_mat( e, f, 1/g );
v1:= m[1][1]*bM[4]+m[1][2]*bM[5];
v2:= m[2][1]*bM[4]+m[2][2]*bM[5];
bM[4]:= v1; bM[5]:= v2;
T:= EmptySCTable( 6, Zero( F ), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,6] );
SetEntrySCTable( T, 4, 5, [1,6] );
N:= LieAlgebraByStructureConstants( F, T );
imgs:= [ ];
c1:= coc21( bM[1],bM[2] );
bM:= Basis( M, bM );
for x in Basis( K ) do
cz:= Coefficients( Basis( C ), x-s( Image( p, x ) ) );
cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) );
Append( cx, cz );
# take care of the coboundary...
cx[6]:= cx[6] - c1*cx[3];
# isomorphism to M_2 ( see paper ):
cx[5]:= cx[5]-cx[2];
Add( imgs, LinearCombination( Basis( N ), cx ) );
od;
return rec( type:= [ 6, 10 ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) );
else
bM[4]:= bM[4]/g;
T:= EmptySCTable( 6, Zero( F ), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,6] );
SetEntrySCTable( T, 4, 5, [1,6] );
N:= LieAlgebraByStructureConstants( F, T );
imgs:= [ ];
c1:= coc21( bM[1],bM[2] );
bM:= Basis( M, bM );
for x in Basis( K ) do
cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) );
cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) );
Append( cx, cz );
# take care of the coboundary...
cx[6]:= cx[6] - c1*cx[3];
Add( imgs, LinearCombination( Basis( N ), cx ) );
od;
return rec( type:= [ 6, 10 ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) );
fi;
elif type = [ 5, 3 ] then
coc21:= function( u, v ) return Coefficients( Basis( C ), coc2( u,v ) )[1]; end;
bM:= ShallowCopy( BasisVectors( Basis( M ) ) );
# we change the basis bM so that the cocycle is in "normal form":
a:= coc21( bM[1],bM[4] );
bM[2]:= bM[2]/a; bM[3]:= bM[3]/a; bM[4]:= bM[4]/a;
b:= coc21( bM[1],bM[5] ); d:= coc21( bM[2],bM[5] );
bM[1]:= bM[1]-b/d*bM[2];
bM[5]:= bM[5]/d;
c:= coc21( bM[2],bM[3] );
if not IsZero( c ) then
T:= EmptySCTable( 6, Zero( F ), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,4] );
SetEntrySCTable( T, 1, 4, [1,6] );
SetEntrySCTable( T, 2, 3, [1,6] );
SetEntrySCTable( T, 2, 5, [1,6] );
N:= LieAlgebraByStructureConstants( F, T );
imgs:= [ ];
c1:= coc21( bM[1],bM[2] );
c2:= coc21( bM[1],bM[3] );
bM:= Basis( M, bM );
for x in Basis( K ) do
cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) );
cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) );
Append( cx, cz );
# take care of the coboundary...
cx[6]:= cx[6] - c1*cx[3]-c2*cx[4];
# make c equal to 1...
cx[1]:= cx[1]/c; cx[2]:= cx[2]/c; cx[3]:= cx[3]/c^2;
cx[4]:= cx[4]/c^3; cx[5]:= cx[5]/c^3; cx[6]:= cx[6]/c^4;
Add( imgs, LinearCombination( Basis( N ), cx ) );
od;
return rec( type:= [ 6, 11 ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) );
else
T:= EmptySCTable( 6, Zero( F ), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,4] );
SetEntrySCTable( T, 1, 4, [1,6] );
SetEntrySCTable( T, 2, 5, [1,6] );
N:= LieAlgebraByStructureConstants( F, T );
imgs:= [ ];
c1:= coc21( bM[1],bM[2] );
c2:= coc21( bM[1],bM[3] );
bM:= Basis( M,bM );
for x in Basis( K ) do
cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) );
cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) );
Append( cx, cz );
# take care of the coboundary...
cx[6]:= cx[6] - c1*cx[3]-c2*cx[4];
Add( imgs, LinearCombination( Basis( N ), cx ) );
od;
return rec( type:= [ 6, 12 ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) );
fi;
elif type = [ 5, 5 ] then
coc21:= function( u, v ) return Coefficients( Basis( C ), coc2( u,v ) )[1]; end;
bM:= ShallowCopy( BasisVectors( Basis( M ) ) );;
# we change the basis bM so that the cocycle is in "normal form":
b:= coc21( bM[1],bM[5] );
bM[2]:= bM[2]/b; bM[3]:= bM[3]/b; bM[5]:= bM[5]/b;
a:= coc21( bM[1],bM[4] ); c:= coc21( bM[2],bM[3] );
bM[1]:= bM[1]-a*bM[3]; bM[2]:= bM[2]+c*bM[4];
if not Characteristic( F )= 2 then
imgs:= [ ];
c1:= coc21( bM[1],bM[2] );
c2:= coc21( bM[1],bM[3] );
d:= coc21( bM[2],bM[4] )-c2;
T:= EmptySCTable( 6, Zero( F ), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,5] );
SetEntrySCTable( T, 2, 4, [1,5] );
SetEntrySCTable( T, 1, 5, [1,6] );
SetEntrySCTable( T, 3, 4, [1,6] );
N:= LieAlgebraByStructureConstants( F, T );
bM:= Basis( M,bM );
for x in Basis( K ) do
cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) );
cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) );
Append( cx, cz );
# take care of the coboundary...
cx[6]:= cx[6] - c1*cx[3]-c2*cx[5];
if not IsZero( d ) then
# map it to 1:
cx[1]:= cx[1]/d; cx[3]:= cx[3]/d; cx[4]:= cx[4]/d^2;
cx[5]:= cx[5]/d^2; cx[6]:= cx[6]/d^3;
# make isom with the Lie alg where d= 0:
cx[5]:= cx[5]+( cx[2]+cx[3] )/2;
cx[4]:= cx[4]+cx[1]/2;
cx[3]:= cx[3]+cx[2];
fi;
Add( imgs, LinearCombination( Basis( N ), cx ) );
od;
return rec( type:= [ 6, 13 ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) );
else
c1:= coc21( bM[1],bM[2] );
c2:= coc21( bM[1],bM[3] );
d:= coc21( bM[2],bM[4] )-c2;
if IsZero( d ) then
imgs:= [ ];
c1:= coc21( bM[1],bM[2] );
c2:= coc21( bM[1],bM[3] );
T:= EmptySCTable( 6, Zero( F ), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,5] );
SetEntrySCTable( T, 2, 4, [1,5] );
SetEntrySCTable( T, 1, 5, [1,6] );
SetEntrySCTable( T, 3, 4, [1,6] );
N:= LieAlgebraByStructureConstants( F, T );
bM:= Basis( M,bM );
for x in Basis( K ) do
cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) );
cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) );
Append( cx, cz );
# take care of the coboundary...
cx[6]:= cx[6] - c1*cx[3]-c2*cx[5];
Add( imgs, LinearCombination( Basis( N ), cx ) );
od;
return rec( type:= [ 6, 13 ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) );
else
imgs:= [ ];
c1:= coc21( bM[1],bM[2] );
c2:= coc21( bM[1],bM[3] );
T:= EmptySCTable( 6, Zero( F ), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,5] );
SetEntrySCTable( T, 1, 5, [1,6] );
SetEntrySCTable( T, 2, 4, [1,5,1,6] );
SetEntrySCTable( T, 3, 4, [1,6] );
N:= LieAlgebraByStructureConstants( F, T );
bM:= Basis( M,bM );
for x in Basis( K ) do
cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) );
cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) );
Append( cx, cz );
# take care of the coboundary...
cx[6]:= cx[6] - c1*cx[3]-c2*cx[5];
cx[1]:= cx[1]/d; cx[3]:= cx[3]/d;
cx[4]:= cx[4]/d^2; cx[5]:= cx[5]/d^2; cx[6]:= cx[6]/d^3;
Add( imgs, LinearCombination( Basis( N ), cx ) );
od;
return rec( type:= [ 6, 29 ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) );
fi;
fi;
elif type = [ 5, 6 ] then
coc21:= function( u, v ) return Coefficients( Basis( C ), coc2( u,v ) )[1]; end;
bM:= ShallowCopy( BasisVectors( Basis( M ) ) );
# we change the basis bM so that the cocycle is in "normal form":
a:= coc21( bM[1],bM[5] );
b:= coc21( bM[2],bM[3] )-coc21( bM[1],bM[4] );
c:= coc21( bM[2],bM[5] );
if not IsZero( c ) then
if not Characteristic( F )= 2 then
a21:= -a/c; a42:= -( a^2/c^2+b/c )/2;
bM[1]:= bM[1]+a21*bM[2];
bM[2]:= bM[2]+a42*bM[4];
bM[3]:= bM[3]+a42*bM[5];
bM[4]:= bM[4]+a21*bM[5];
c:= coc21( bM[2],bM[5] );
imgs:= [ ];
c1:= coc21( bM[1],bM[2] );
c2:= coc21( bM[1],bM[3] );
c3:= coc21( bM[1],bM[4] );
T:= EmptySCTable( 6, Zero( F ), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,4] );
SetEntrySCTable( T, 1, 4, [1,5] );
SetEntrySCTable( T, 2, 3, [1,5] );
SetEntrySCTable( T, 2, 5, [1,6] );
SetEntrySCTable( T, 3, 4, [-1,6] );
N:= LieAlgebraByStructureConstants( F, T );
bM:= Basis( M, bM );
for x in Basis( K ) do
cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) );
cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) );
Append( cx, cz );
# take care of the coboundary...
cx[6]:= cx[6] - c1*cx[3]-c2*cx[4]-c3*cx[5];
# make c= 1;
cx[6]:= cx[6]/c;
Add( imgs, LinearCombination( Basis( N ), cx ) );
od;
return rec( type:= [ 6, 14 ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) );
elif Characteristic( F )= 2 and IsFinite(F) then
a21:= a/c; a32:= a/c+( b/c )^( Size( F )/2 );
bM[1]:= bM[1]+a21*bM[2];
bM[2]:= bM[2]+a32*bM[3];
bM[3]:= bM[3]+a32*bM[4]+a21*a32*bM[5];
bM[4]:= bM[4]+( a21+a32 )*bM[5];
c:= coc21( bM[2],bM[5] );
imgs:= [ ];
c1:= coc21( bM[1],bM[2] );
c2:= coc21( bM[1],bM[3] );
c3:= coc21( bM[1],bM[4] );
T:= EmptySCTable( 6, Zero( F ), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,4] );
SetEntrySCTable( T, 1, 4, [1,5] );
SetEntrySCTable( T, 2, 3, [1,5] );
SetEntrySCTable( T, 2, 5, [1,6] );
SetEntrySCTable( T, 3, 4, [1,6] );
N:= LieAlgebraByStructureConstants( F, T );
bM:= Basis( M, bM );
for x in Basis( K ) do
cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) );
cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) );
Append( cx, cz );
# take care of the coboundary...
cx[6]:= cx[6] - c1*cx[3]-c2*cx[4]-c3*cx[5];
# make c= 1;
cx[6]:= cx[6]/c;
Add( imgs, LinearCombination( Basis( N ), cx ) );
od;
return rec( type:= [ 6, 14 ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) );
else
a21:= a/c; a32:= a/c;
bM[1]:= bM[1]+a21*bM[2];
bM[2]:= bM[2]+a32*bM[3];
bM[3]:= bM[3]+a32*bM[4]+a21*a32*bM[5];
bM[4]:= bM[4]+( a21+a32 )*bM[5];
c:= coc21( bM[2],bM[5] );
imgs:= [ ];
c1:= coc21( bM[1],bM[2] );
c2:= coc21( bM[1],bM[3] );
c3:= coc21( bM[1],bM[4] );
T:= EmptySCTable( 6, Zero( F ), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,4] );
SetEntrySCTable( T, 1, 4, [1,5] );
SetEntrySCTable( T, 2, 3, [1,5,b/c,6] );
SetEntrySCTable( T, 2, 5, [1,6] );
SetEntrySCTable( T, 3, 4, [1,6] );
N:= LieAlgebraByStructureConstants( F, T );
bM:= Basis( M, bM );
for x in Basis( K ) do
cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) );
cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) );
Append( cx, cz );
# take care of the coboundary...
cx[6]:= cx[6] - c1*cx[3]-c2*cx[4]-c3*cx[5];
# make c= 1;
cx[6]:= cx[6]/c;
Add( imgs, LinearCombination( Basis( N ), cx ) );
od;
return rec( type:= [ 6, 31, b/c ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) );
fi;
else # i.e., c = 0
if not Characteristic( F )= 2 then
a21:= b/( 2*a );
bM[1]:= bM[1]+a21*bM[2];
bM[4]:= bM[4]+a21*bM[5];
imgs:= [ ];
c1:= coc21( bM[1],bM[2] );
c2:= coc21( bM[1],bM[3] );
c3:= coc21( bM[1],bM[4] );
T:= EmptySCTable( 6, Zero( F ), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,4] );
SetEntrySCTable( T, 1, 4, [1,5] );
SetEntrySCTable( T, 2, 3, [1,5] );
SetEntrySCTable( T, 1, 5, [1,6] );
SetEntrySCTable( T, 2, 4, [1,6] );
N:= LieAlgebraByStructureConstants( F, T );
bM:= Basis( M, bM );
for x in Basis( K ) do
cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) );
cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) );
Append( cx, cz );
# take care of the coboundary...
cx[6]:= cx[6] - c1*cx[3]-c2*cx[4]-c3*cx[5];
# make a= 1:
cx[6]:= cx[6]/a;
Add( imgs, LinearCombination( Basis( N ), cx ) );
od;
return rec( type:= [ 6, 15 ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) );
else
imgs:= [ ];
c1:= coc21( bM[1],bM[2] );
c2:= coc21( bM[1],bM[3] );
c3:= coc21( bM[1],bM[4] );
if not IsZero( b ) then
T:= EmptySCTable( 6, Zero( F ), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,4] );
SetEntrySCTable( T, 1, 4, [1,5] );
SetEntrySCTable( T, 1, 5, [1,6] );
SetEntrySCTable( T, 2, 3, [1,5,1,6] );
SetEntrySCTable( T, 2, 4, [1,6] );
N:= LieAlgebraByStructureConstants( F, T );
tt:= [ 6, 30 ];
else
T:= EmptySCTable( 6, Zero( F ), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,4] );
SetEntrySCTable( T, 1, 4, [1,5] );
SetEntrySCTable( T, 1, 5, [1,6] );
SetEntrySCTable( T, 2, 3, [1,5] );
SetEntrySCTable( T, 2, 4, [1,6] );
N:= LieAlgebraByStructureConstants( F, T );
tt:= [ 6, 15 ];
fi;
bM:= Basis( M, bM );
b:= b/a; #( we will divide by a, also changing b )
for x in Basis( K ) do
cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) );
cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) );
Append( cx, cz );
# take care of the coboundary...
cx[6]:= cx[6] - c1*cx[3]-c2*cx[4]-c3*cx[5];
# make a= 1:
cx[6]:= cx[6]/a;
if not IsZero( b ) then
# make it 1:
cx[1]:= cx[1]/b; cx[2]:= cx[2]/b^2; cx[3]:= cx[3]/b^3;
cx[4]:= cx[4]/b^4; cx[5]:= cx[5]/b^5; cx[6]:= cx[6]/b^6;
fi;
Add( imgs, LinearCombination( Basis( N ), cx ) );
od;
return rec( type:= tt, f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) );
fi;
fi;
elif type = [ 5, 7 ] then
coc21:= function( u, v ) return Coefficients( Basis( C ), coc2( u,v ) )[1]; end;
bM:= ShallowCopy( BasisVectors( Basis( M ) ) );
# we change the basis bM so that the cocycle is in "normal form":
a:= coc21( bM[1],bM[5] );
b:= coc21( bM[2],bM[3] );
c:= coc21( bM[2],bM[5] );
if not IsZero( c ) then
a21:= -a/c;
bM[1]:= bM[1]+a21*bM[2];
if not Characteristic( F )= 2 then
a42:= -( b/c )/2;
bM[2]:= bM[2]+a42*bM[4];
bM[3]:= bM[3]+a42*bM[5];
c:= coc21( bM[2],bM[5] );
imgs:= [ ];
c1:= coc21( bM[1],bM[2] );
c2:= coc21( bM[1],bM[3] );
c3:= coc21( bM[1],bM[4] );
T:= EmptySCTable( 6, Zero( F ), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,4] );
SetEntrySCTable( T, 1, 4, [1,5] );
SetEntrySCTable( T, 2, 5, [1,6] );
SetEntrySCTable( T, 3, 4, [-1,6] );
N:= LieAlgebraByStructureConstants( F, T );
bM:= Basis( M, bM );
for x in Basis( K ) do
cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) );
cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) );
Append( cx, cz );
# take care of the coboundary...
cx[6]:= cx[6] - c1*cx[3]-c2*cx[4]-c3*cx[5];
# make c= 1;
cx[6]:= cx[6]/c;
Add( imgs, LinearCombination( Basis( N ), cx ) );
od;
return rec( type:= [ 6, 16 ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) );
elif Characteristic( F )= 2 and IsFinite(F) then
a32:= ( b/c )^( Size( F )/2 );
bM[2]:= bM[2]+a32*bM[3];
bM[3]:= bM[3]+a32*bM[4];
bM[4]:= bM[4]+a32*bM[5];
c:= coc21( bM[2],bM[5] );
imgs:= [ ];
c1:= coc21( bM[1],bM[2] );
c2:= coc21( bM[1],bM[3] );
c3:= coc21( bM[1],bM[4] );
T:= EmptySCTable( 6, Zero( F ), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,4] );
SetEntrySCTable( T, 1, 4, [1,5] );
SetEntrySCTable( T, 2, 5, [1,6] );
SetEntrySCTable( T, 3, 4, [1,6] );
N:= LieAlgebraByStructureConstants( F, T );
bM:= Basis( M, bM );
for x in Basis( K ) do
cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) );
cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) );
Append( cx, cz );
# take care of the coboundary...
cx[6]:= cx[6] - c1*cx[3]-c2*cx[4]-c3*cx[5];
# make c= 1;
cx[6]:= cx[6]/c;
Add( imgs, LinearCombination( Basis( N ), cx ) );
od;
return rec( type:= [ 6, 16 ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) );
else
c:= coc21( bM[2],bM[5] );
imgs:= [ ];
c1:= coc21( bM[1],bM[2] );
c2:= coc21( bM[1],bM[3] );
c3:= coc21( bM[1],bM[4] );
T:= EmptySCTable( 6, Zero( F ), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,4] );
SetEntrySCTable( T, 1, 4, [1,5] );
SetEntrySCTable( T, 2, 3, [b/c,6] );
SetEntrySCTable( T, 2, 5, [1,6] );
SetEntrySCTable( T, 3, 4, [1,6] );
N:= LieAlgebraByStructureConstants( F, T );
bM:= Basis( M, bM );
for x in Basis( K ) do
cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) );
cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) );
Append( cx, cz );
# take care of the coboundary...
cx[6]:= cx[6] - c1*cx[3]-c2*cx[4]-c3*cx[5];
# make c= 1;
cx[6]:= cx[6]/c;
Add( imgs, LinearCombination( Basis( N ), cx ) );
od;
return rec( type:= [ 6, 32, b/c ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) );
fi;
else
imgs:= [ ];
c1:= coc21( bM[1],bM[2] );
c2:= coc21( bM[1],bM[3] );
c3:= coc21( bM[1],bM[4] );
if not IsZero( b ) then
T:= EmptySCTable( 6, Zero( F ), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,4] );
SetEntrySCTable( T, 1, 4, [1,5] );
SetEntrySCTable( T, 1, 5, [1,6] );
SetEntrySCTable( T, 2, 3, [1,6] );
N:= LieAlgebraByStructureConstants( F, T );
tt:= [ 6, 17 ];
else
T:= EmptySCTable( 6, Zero( F ), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,4] );
SetEntrySCTable( T, 1, 4, [1,5] );
SetEntrySCTable( T, 1, 5, [1,6] );
N:= LieAlgebraByStructureConstants( F, T );
tt:= [ 6, 18 ];
fi;
bM:= Basis( M, bM );
b:= b/a; #( we will divide by a, also changing b )
for x in Basis( K ) do
cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) );
cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) );
Append( cx, cz );
# take care of the coboundary...
cx[6]:= cx[6] - c1*cx[3]-c2*cx[4]-c3*cx[5];
# make a= 1:
cx[6]:= cx[6]/a;
if not IsZero( b ) then
# make it 1:
cx[1]:= cx[1]/b; cx[2]:= cx[2]/b^2; cx[3]:= cx[3]/b^3;
cx[4]:= cx[4]/b^4; cx[5]:= cx[5]/b^5; cx[6]:= cx[6]/b^6;
fi;
Add( imgs, LinearCombination( Basis( N ), cx ) );
od;
return rec( type:= tt, f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) );
fi;
elif type = [ 5, 8 ] then
coc21:= function( u, v ) return Coefficients( Basis( C ), coc2( u,v ) )[1]; end;
bM:= ShallowCopy( BasisVectors( Basis( M ) ) );
# we change the basis bM so that the cocycle is in "normal form":
a:= coc21( bM[1],bM[4] );
b:= coc21( bM[1],bM[5] );
c:= coc21( bM[2],bM[3] );
d:= coc21( bM[2],bM[4] );
e:= coc21( bM[2],bM[5] );
f:= coc21( bM[3],bM[5] );
if not IsZero( f ) then
bM[2]:= bM[2]-e/f*bM[3];
bM[4]:= bM[4]-e/f*bM[5];
elif not IsZero( d ) then
bM[3]:= bM[3]-e/d*bM[2];
bM[5]:= bM[5]-e/d*bM[4];
else
if not Characteristic( F )= 2 then
v:= bM[2];
bM[2]:= bM[2]-bM[3];
bM[3]:= bM[3]+v;
v:= bM[4];
bM[4]:= bM[4]-bM[5];
bM[5]:= v+bM[5];
fi;
fi;
d:= coc21( bM[2],bM[4] );
if not IsZero( d ) then
# make d= 1:
bM[1]:= bM[1]/d; bM[4]:= bM[4]/d; bM[5]:= bM[5]/d;
# and a= c= 0
a:= coc21( bM[1],bM[4] );
c:= coc21( bM[2],bM[3] );
bM[1]:= bM[1]-a*bM[2];
bM[3]:= bM[3]-c*bM[4];
b:= coc21( bM[1],bM[5] );
if not IsZero( b ) then
bM[3]:= bM[3]/b;
bM[5]:= bM[5]/b;
f:= coc21( bM[3],bM[5] );
if not IsZero( f ) then
if IsFinite( F ) then
if not Characteristic( F )= 2 then
xi:= PrimitiveElement( F );
ex:= LogFFE( f, xi );
if IsEvenInt( ex ) then
f:= One( F );
eps:= xi^( ex/2 );
else
f:= xi;
eps:= xi^( ( ex-1 )/2 );
fi;
else
eps:= f^( Size( F )/2 );
f:= One( F );
fi;
elif F = Rationals then
u:= LieAlgDBHelper.sq_free( f );
eps:= u[1];
f:= u[2];
else
eps:= 1;
fi;
T:= EmptySCTable( 6, Zero( F ), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,4] );
SetEntrySCTable( T, 1, 3, [1,5] );
SetEntrySCTable( T, 1, 5, [1,6] );
SetEntrySCTable( T, 2, 4, [1,6] );
SetEntrySCTable( T, 3, 5, [f,6] );
N:= LieAlgebraByStructureConstants( F, T );
bM:= Basis( M, bM );
imgs:= [ ];
c1:= coc21( bM[1],bM[2] );
c2:= coc21( bM[1],bM[3] );
for x in Basis( K ) do
cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) );
cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) );
Append( cx, cz );
# take care of the coboundary...
cx[6]:= cx[6] - c1*cx[4]-c2*cx[5];
# isom
cx[2]:= cx[2]*eps;
cx[3]:= cx[3]*eps^2;
cx[4]:= cx[4]*eps;
cx[5]:= cx[5]*eps^2;
cx[6]:= cx[6]*eps^2;
Add( imgs, LinearCombination( Basis( N ), cx ) );
od;
return rec( type:= [ 6, 19, f ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) );
else # i.e., f= 0
T:= EmptySCTable( 6, Zero( F ), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,4] );
SetEntrySCTable( T, 1, 3, [1,5] );
SetEntrySCTable( T, 1, 5, [1,6] );
SetEntrySCTable( T, 2, 4, [1,6] );
N:= LieAlgebraByStructureConstants( F, T );
bM:= Basis( M, bM );
imgs:= [ ];
c1:= coc21( bM[1],bM[2] );
c2:= coc21( bM[1],bM[3] );
for x in Basis( K ) do
cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) );
cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) );
Append( cx, cz );
# take care of the coboundary...
cx[6]:= cx[6] - c1*cx[4]-c2*cx[5];
Add( imgs, LinearCombination( Basis( N ), cx ) );
od;
return rec( type:= [ 6, 20 ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) );
fi;
else # i.e., b = 0
f:= coc21( bM[3],bM[5] );
if not IsZero( f ) then
ff:= ShallowCopy( f );
if IsFinite( F ) then
if not Characteristic( F )= 2 then
xi:= PrimitiveElement( F );
ex:= LogFFE( f, xi );
if IsEvenInt( ex ) then
f:= One( F );
eps:= xi^( ex/2 );
else
f:= xi;
eps:= xi^( ( ex-1 )/2 );
fi;
else
eps:= f^( Size( F )/2 );
f:= One( F );
fi;
elif F = Rationals then
u:= LieAlgDBHelper.sq_free( f );
eps:= u[1];
f:= u[2];
else
eps:= 1;
fi;
T:= EmptySCTable( 6, Zero( F ), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,4] );
SetEntrySCTable( T, 1, 3, [1,5] );
SetEntrySCTable( T, 1, 5, [1,6] );
SetEntrySCTable( T, 2, 4, [1,6] );
SetEntrySCTable( T, 3, 5, [f,6] );
N:= LieAlgebraByStructureConstants( F, T );
bM:= Basis( M, bM );
imgs:= [ ];
c1:= coc21( bM[1],bM[2] );
c2:= coc21( bM[1],bM[3] );
for x in Basis( K ) do
cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) );
cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) );
Append( cx, cz );
cx[3]:= cx[3] - cx[1]/ff;
# take care of the coboundary...
cx[6]:= cx[6] - c1*cx[4]-c2*cx[5];
# isom
cx[2]:= cx[2]*eps;
cx[3]:= cx[3]*eps^2;
cx[4]:= cx[4]*eps;
cx[5]:= cx[5]*eps^2;
cx[6]:= cx[6]*eps^2;
Add( imgs, LinearCombination( Basis( N ), cx ) );
od;
return rec( type:= [ 6, 19, f ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) );
fi;
fi;
else # i.e., d= 0
f:= coc21( bM[3],bM[5] );
if not IsZero( f ) then
# make f= 1:
bM[1]:= bM[1]/f; bM[4]:= bM[4]/f; bM[5]:= bM[5]/f;
# and c= 0, a= 1
a:= coc21( bM[1],bM[4] );
c:= coc21( bM[2],bM[3] );
bM[2]:= bM[2]+c*bM[5];
bM[2]:= bM[2]/a; bM[4]:= bM[4]/a;
b:= coc21( bM[1],bM[5] );
T:= EmptySCTable( 6, Zero( F ), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,4] );
SetEntrySCTable( T, 1, 3, [1,5] );
SetEntrySCTable( T, 1, 5, [1,6] );
SetEntrySCTable( T, 2, 4, [1,6] );
N:= LieAlgebraByStructureConstants( F, T );
bM:= Basis( M, bM );
imgs:= [ ];
c1:= coc21( bM[1],bM[2] );
c2:= coc21( bM[1],bM[3] );
for x in Basis( K ) do
cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) );
cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) );
Append( cx, cz );
# take care of the coboundary...
cx[6]:= cx[6] - c1*cx[4]-c2*cx[5];
if not IsZero( b ) then
cx[2]:= cx[2]/b^2; cx[3]:= cx[3]/b; cx[4]:= cx[4]/b^2;
cx[5]:= cx[5]/b; cx[6]:= cx[6]/b^2;
cx[3]:= cx[3]+cx[1];
fi;
cx1:= ShallowCopy( cx );
cx1[2]:= cx[3]; cx1[3]:= cx[2];
cx1[4]:= cx[5]; cx1[5]:= cx[4];
Add( imgs, LinearCombination( Basis( N ), cx1 ) );
od;
return rec( type:= [ 6, 20 ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) );
else # i.d., d= f= 0 and char 2
# make e= 1:
bM[1]:= bM[1]/e; bM[4]:= bM[4]/e; bM[5]:= bM[5]/e;
# and a= b= c= 0,
a:= coc21( bM[1],bM[4] );
b:= coc21( bM[1],bM[5] );
bM[1]:= bM[1]-b*bM[2]-a*bM[3];
bM[3]:= bM[3]-c*bM[5];
T:= EmptySCTable( 6, Zero( F ), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,4] );
SetEntrySCTable( T, 1, 3, [1,5] );
SetEntrySCTable( T, 2, 5, [1,6] );
SetEntrySCTable( T, 3, 4, [1,6] );
N:= LieAlgebraByStructureConstants( F, T );
bM:= Basis( M, bM );
imgs:= [ ];
c1:= coc21( bM[1],bM[2] );
c2:= coc21( bM[1],bM[3] );
for x in Basis( K ) do
cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) );
cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) );
Append( cx, cz );
cx[6]:= cx[6]-c1*cx[4]-c2*cx[5];
Add( imgs, LinearCombination( Basis( N ), cx ) );
od;
return rec( type:= [ 6, 33 ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) );
fi;
fi;
elif type = [ 5, 9 ] then
coc21:= function( u, v ) return Coefficients( Basis( C ), coc2( u,v ) )[1]; end;
bM:= ShallowCopy( BasisVectors( Basis( M ) ) );
# we change the basis bM so that the cocycle is in "normal form":
a:= coc21( bM[1],bM[4] );
b:= coc21( bM[1],bM[5] );
c:= coc21( bM[2],bM[5] );
if not IsZero( c ) then
bM[1]:= bM[1]-b/c*bM[2];
bM[4]:= bM[4]-b/c*bM[5];
elif not IsZero( a ) then
bM[2]:= bM[2]-b/a*bM[1];
bM[5]:= bM[5]-b/a*bM[4];
else
if not Characteristic( F )= 2 then
v:= bM[1];
bM[1]:= bM[1]-bM[2];
bM[2]:= v+bM[2];
bM[3]:= 2*bM[3];
v:= bM[4];
bM[4]:= 2*( bM[4]-bM[5] );
bM[5]:= 2*( v+bM[5] );
fi;
fi;
b:= coc21( bM[1],bM[5] );
if IsZero( b ) then
a:= coc21( bM[1],bM[4] );
c:= coc21( bM[2],bM[5] );
aa:= ShallowCopy( a );
if IsFinite( F ) then
if not Characteristic( F )= 2 then
xi:= PrimitiveElement( F );
ex:= LogFFE( c/a, xi );
if IsEvenInt( ex ) then
c:= One( F )*a;
eps:= xi^( ex/2 );
else
c:= xi*a;
eps:= xi^( ( ex-1 )/2 );
fi;
else
eps:= ( c/a )^( Size( F )/2 );
c:= One( F )*a;
fi;
elif F = Rationals then
u:= LieAlgDBHelper.sq_free( c/a );
eps:= u[1];
c:= u[2]*a;
else
eps:= 1;
fi;
T:= EmptySCTable( 6, Zero( F ), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,4] );
SetEntrySCTable( T, 2, 3, [1,5] );
SetEntrySCTable( T, 1, 4, [1,6] );
SetEntrySCTable( T, 2, 5, [c/a,6] );
N:= LieAlgebraByStructureConstants( F, T );
bM:= Basis( M, bM );
imgs:= [ ];
c1:= coc21( bM[1],bM[2] );
c2:= coc21( bM[1],bM[3] );
c3:= coc21( bM[2],bM[3] );
for x in Basis( K ) do
cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) );
cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) );
Append( cx, cz );
# take care of the coboundary...
cx[6]:= cx[6] - c1*cx[3]-c2*cx[4]-c3*cx[5];
cx[6]:= cx[6]/aa;
# isom
cx[2]:= cx[2]*eps;
cx[3]:= cx[3]*eps;
cx[4]:= cx[4]*eps;
cx[5]:= cx[5]*eps^2;
cx[6]:= cx[6]*eps;
Add( imgs, LinearCombination( Basis( N ), cx ) );
od;
return rec( type:= [ 6, 21, c/a ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) );
else
T:= EmptySCTable( 6, Zero( F ), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,4] );
SetEntrySCTable( T, 1, 5, [1,6] );
SetEntrySCTable( T, 2, 3, [1,5] );
SetEntrySCTable( T, 2, 4, [1,6] );
N:= LieAlgebraByStructureConstants( F, T );
bM:= Basis( M, bM );
imgs:= [ ];
c1:= coc21( bM[1],bM[2] );
c2:= coc21( bM[1],bM[3] );
c3:= coc21( bM[2],bM[3] );
for x in Basis( K ) do
cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) );
cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) );
Append( cx, cz );
# take care of the coboundary...
cx[6]:= cx[6] - c1*cx[3]-c2*cx[4]-c3*cx[5];
cx[6]:= cx[6]/b;
Add( imgs, LinearCombination( Basis( N ), cx ) );
od;
return rec( type:= [ 6, 34 ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) );
fi;
elif type = [ 4, 1 ] then
coc21:= function( u, v ) return Coefficients( Basis( C ), coc2( u,v ) )[1]; end;
coc22:= function( u, v ) return Coefficients( Basis( C ), coc2( u,v ) )[2]; end;
bM:= LieAlgDBHelper.skew_symm_NF( M, coc21 );
if IsZero( coc21( bM[3],bM[4] ) ) then
# find a linear combination of coc21, coc22 that is nondegenerate:
y:= coc22( bM[1],bM[2] ); a:= coc22( bM[1],bM[3] );
b:= coc22( bM[1],bM[4] ); c:= coc22( bM[2],bM[3] );
d:= coc22( bM[2],bM[4] ); e:= coc22( bM[3],bM[4] );
if IsZero( a*d-b*c ) then
x:= 1-y;
else
x:= -y;
fi;
if IsZero( x ) then
store:= coc21;
coc21:= coc22;
# so now coc21 has become coc22, need to make coc22 equal to
# the old coc21 ( otherwise no longer independent ).
coc22:= store;
r:= 1; t:= 0;
else
f1:= coc21;
f2:= ShallowCopy( coc22 );
coc21:= function( u,v ) return x*f1( u,v )+f2( u,v ); end;
r:= 0; t:= 1;
fi;
bM:= LieAlgDBHelper.skew_symm_NF( M, coc21 );
u:= x; v:= 1;
# i.e., we have made a base change in the centre, corresponding
# to theta_1 --> u*theta_1 + v*\theta_2
# theta_2 --> r*theta_1 + t*\theta_2
else
u:= 1; v:= 0;
r:= 0; t:= 1;
fi;
y:= coc22( bM[1],bM[2] );
if not IsZero( y ) then
f2:= coc22;
coc22:= function( u,v ) return -y*coc21( u,v )+f2( u,v ); end;
r:= -u*y+r; t:= -v*y+t;
# a new base change...
fi;
a:= coc22( bM[1],bM[3] );
if IsZero( a ) then
# try to make it not 0:
b:= coc22( bM[1],bM[4] );
c:= coc22( bM[2],bM[3] );
d:= coc22( bM[2],bM[4] );
if not IsZero( b ) then
bM[3]:= bM[3]+bM[4];
a:= coc22( bM[1],bM[3] );
elif not IsZero( c ) then
bM[1]:= bM[1]+bM[2];
a:= coc22( bM[1],bM[3] );
elif not IsZero( d ) then
bM[1]:= bM[1]+bM[2];
bM[3]:= bM[3]+bM[4];
a:= coc22( bM[1],bM[3] );
fi;
fi;
if not IsZero( a ) then
# make it 1:
bM[1]:= bM[1]/a; bM[4]:= bM[4]/a;
# make b,c -> 0
b:= coc22( bM[1],bM[4] ); c:= coc22( bM[2],bM[3] );
bM[2]:= bM[2]-c*bM[1]; bM[4]:= bM[4]-b*bM[3];
e:= coc22( bM[3],bM[4] );
if IsZero( e ) then
b:= coc22( bM[2],bM[4] );
bb:= ShallowCopy( b );
if not IsZero( b ) then
if IsFinite( F ) then
if not Characteristic( F )= 2 then
xi:= PrimitiveElement( F );
ex:= LogFFE( b, xi );
if IsEvenInt( ex ) then
b:= One( F );
eps:= xi^( ex/2 );
else
b:= xi;
eps:= xi^( ( ex-1 )/2 );
fi;
else
eps:= b^( Size( F )/2 );
b:= Zero( F );
fi;
elif F = Rationals then
uu:= LieAlgDBHelper.sq_free( b );
eps:= uu[1];
b:= uu[2];
else
eps:= 1;
fi;
fi;
T:= EmptySCTable( 6, Zero( F ), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,5] );
SetEntrySCTable( T, 1, 3, [1,6] );
SetEntrySCTable( T, 2, 4, [b,6] );
SetEntrySCTable( T, 3, 4, [1,5] );
N:= LieAlgebraByStructureConstants( F, T );
bM:= Basis( M, bM );
imgs:= [ ];
q:= coc21( bM[1],bM[2] ); # we have to change that back to 1...
for x in Basis( K ) do
cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) );
cz1:= ShallowCopy( cz );
cz1[1]:= u/q*cz[1]+v/q*cz[2];
cz1[2]:= r*cz[1]+t*cz[2];
cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) );
Append( cx, cz1 );
if not IsZero( bb ) then
# isom
cx[1]:= cx[1]/eps;
cx[3]:= cx[3]/eps;
cx[5]:= cx[5]/eps;
cx[6]:= cx[6]/eps^2;
if Characteristic( F )= 2 then
# isom
cx1:= ShallowCopy( cx );
cx1[1]:= cx[1]+cx[4];
cx1[2]:= cx[2];
cx1[3]:= cx[2]+cx[3];
cx1[4]:= cx[1];
cx1[5]:= cx[6];
cx1[6]:= cx[5]+cx[6];
cx:= ShallowCopy( cx1 );
fi;
fi;
Add( imgs, LinearCombination( Basis( N ), cx ) );
od;
return rec( type:= [ 6, 22, b ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) );
else
# make it 1:
bM[2]:= bM[2]/e; bM[4]:= bM[4]/e;
a:= coc22( bM[2],bM[4] );
if not Characteristic( F )= 2 then
if not IsZero( a ) and not IsZero( a+1/4 ) then
b:= a+1/4;
elif IsZero( a ) then
b:= One( F );
elif IsZero( a+1/4 ) then
b:= Zero( F );
fi;
bb:= ShallowCopy( b );
if not IsZero( b ) then
if IsFinite( F ) then
xi:= PrimitiveElement( F );
ex:= LogFFE( b, xi );
if IsEvenInt( ex ) then
b:= One( F );
eps:= xi^( ex/2 );
else
b:= xi;
eps:= xi^( ( ex-1 )/2 );
fi;
elif F = Rationals then
uu:= LieAlgDBHelper.sq_free( b );
eps:= uu[1];
b:= uu[2];
else
eps:= 1;
fi;
fi;
T:= EmptySCTable( 6, Zero( F ), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,5] );
SetEntrySCTable( T, 1, 3, [1,6] );
SetEntrySCTable( T, 2, 4, [b,6] );
SetEntrySCTable( T, 3, 4, [1,5] );
N:= LieAlgebraByStructureConstants( F, T );
bM:= Basis( M, bM );
imgs:= [ ];
q:= coc21( bM[1],bM[2] ); # we have to change that back to 1...
for x in Basis( K ) do
cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) );
cz1:= ShallowCopy( cz );
cz1[1]:= u/q*cz[1]+v/q*cz[2];
cz1[2]:= r*cz[1]+t*cz[2];
cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) );
Append( cx, cz1 );
cx1:= ShallowCopy( cx );
if not IsZero( a ) and not IsZero( a+1/4 ) then
cx1[1]:= ( 1/( 4*a )+1 )*cx[1];
cx1[2]:= cx[3];
cx1[3]:= a*cx[2]+cx[3]/2;
cx1[4]:= cx[4]+1/( 2*a )*cx[1];
cx1[5]:= -cx[5]/2+cx[6];
cx1[6]:= ( a+1/4 )*cx[5];
elif IsZero( a ) then
cx1[1]:= cx[4];
cx1[2]:= -cx[2]/4;
cx1[3]:= cx[2]/4-cx[3]/2;
cx1[4]:= 2*cx[1]-cx[4];
cx1[5]:= -cx[5]/2+cx[6];
cx1[6]:= cx[5]/2;
elif IsZero( a+1/4 ) then
cx1[1]:= -2*cx[1]+cx[4];
cx1[2]:= cx[2]/2-2*cx[3];
cx1[3]:= cx[2]/2-cx[3];
cx1[4]:= -4*cx[1]+cx[4];
cx1[6]:= -cx[5]+2*cx[6];
fi;
cx:= ShallowCopy( cx1 );
if not IsZero( bb ) then
# isom
cx[1]:= cx[1]/eps;
cx[3]:= cx[3]/eps;
cx[5]:= cx[5]/eps;
cx[6]:= cx[6]/eps^2;
fi;
Add( imgs, LinearCombination( Basis( N ), cx ) );
od;
return rec( type:= [ 6, 22, b ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) );
else
b:= coc22( bM[2],bM[4] );
bb:= ShallowCopy( b );
if not IsZero( b ) then
if IsFinite( F ) then
ex:= First( [0..( Size( F )-1 )],i->Trace( F,PrimitiveElement( F )^i )= One( F ) );
xi:= PrimitiveElement( F )^ex;
P:= PolynomialRing( F,["y"] );
y:= P.1;
if Trace( F, b )= Zero( F ) then
root:= Factors( P,y^2+y+b )[1];
eps:= CoefficientsOfUnivariatePolynomial( root )[1];
b:= Zero( F );
else
root:= Factors( P,y^2+y+b+xi )[1];
eps:= CoefficientsOfUnivariatePolynomial( root )[1];
b:= xi;
fi;
else
eps:= 1;
fi;
fi;
T:= EmptySCTable( 6, Zero( F ), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,5] );
SetEntrySCTable( T, 1, 3, [1,6] );
SetEntrySCTable( T, 2, 4, [b,6] );
SetEntrySCTable( T, 3, 4, [1,5,1,6] );
N:= LieAlgebraByStructureConstants( F, T );
bM:= Basis( M, bM );
imgs:= [ ];
q:= coc21( bM[1],bM[2] ); # we have to change that back to 1...
for x in Basis( K ) do
cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) );
cz1:= ShallowCopy( cz );
cz1[1]:= u/q*cz[1]+v/q*cz[2];
cz1[2]:= r*cz[1]+t*cz[2];
cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) );
Append( cx, cz1 );
if not IsZero( bb ) then
# isom
cx1:= ShallowCopy( cx );
cx1[1]:= cx[1]+cx[4]*eps;
cx1[2]:= cx[2];
cx1[3]:= cx[2]*eps+cx[3];
cx1[4]:= cx[4];
cx1[5]:= cx[5];
cx1[6]:= cx[5]*eps+cx[6];
cx:= ShallowCopy( cx1 );
fi;
Add( imgs, LinearCombination( Basis( N ), cx ) );
od;
return rec( type:= [ 6, 35, b ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) );
fi;
fi;
else # i.e, a= 0, which means that b= c= d= 0
# make e= 1:
e:= coc22( bM[3],bM[4] );
f2:= coc22;
coc22:= function( u,v ) return f2( u,v )/e; end;
r:= r/e; t:= t/e;
# subtract coc22 from coc21:
f1:= coc21;
coc21:= function( u,v ) return f1( u,v )-coc22( u,v ); end;
u:= u-r; v:= v-t;
if not Characteristic( F )= 2 then
T:= EmptySCTable( 6, Zero( F ), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,5] );
SetEntrySCTable( T, 1, 3, [1,6] );
SetEntrySCTable( T, 2, 4, [1,6] );
SetEntrySCTable( T, 3, 4, [1,5] );
N:= LieAlgebraByStructureConstants( F, T );
bM:= Basis( M, bM );
imgs:= [ ];
for x in Basis( K ) do
cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) );
cz1:= ShallowCopy( cz );
cz1[1]:= u*cz[1]+v*cz[2];
cz1[2]:= r*cz[1]+t*cz[2];
cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) );
Append( cx, cz1 );
# isom
cx1:= ShallowCopy( cx );
cx1[1]:= cx[1]-cx[4];
cx1[2]:= cx[2]/4-cx[3]/4;
cx1[3]:= cx[2]/4+cx[3]/4;
cx1[4]:= -cx[1]-cx[4];
cx1[5]:= cx[5]/2-cx[6]/2;
cx1[6]:= cx[5]/2+cx[6]/2;
cx:= ShallowCopy( cx1 );
Add( imgs, LinearCombination( Basis( N ), cx ) );
od;
return rec( type:= [ 6, 22, One( F ) ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) );
else
T:= EmptySCTable( 6, Zero( F ), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,5] );
SetEntrySCTable( T, 1, 3, [1,6] );
SetEntrySCTable( T, 3, 4, [1,5,1,6] );
N:= LieAlgebraByStructureConstants( F, T );
bM:= Basis( M, bM );
imgs:= [ ];
for x in Basis( K ) do
cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) );
cz1:= ShallowCopy( cz );
cz1[1]:= u*cz[1]+v*cz[2];
cz1[2]:= r*cz[1]+t*cz[2];
cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) );
Append( cx, cz1 );
# isom
cx1:= ShallowCopy( cx );
cx1[1]:= cx[4];
cx1[2]:= cx[2]+cx[3];
cx1[3]:= cx[2];
cx1[4]:= cx[1]+cx[4];
cx1[5]:= cx[5]+cx[6];
cx1[6]:= cx[5];
cx:= ShallowCopy( cx1 );
Add( imgs, LinearCombination( Basis( N ), cx ) );
od;
return rec( type:= [ 6, 35, Zero( F ) ], f:= LieAlgDBHelper.liealg_hom( K, N, Basis( K ), imgs ) );
fi;
fi;
elif type = [ 4, 2 ] then
coc21:= function( u, v ) return Coefficients( Basis( C ), coc2( u,v ) )[1]; end;
coc22:= function( u, v ) return Coefficients( Basis( C ), coc2( u,v ) )[2]; end;
bM:= ShallowCopy( BasisVectors( Basis( M ) ) );
# coc21 can be moved to one of three normal forms
a:= coc21( bM[1],bM[3] );
b:= coc21( bM[1],bM[4] );
c:= coc21( bM[2],bM[3] );
d:= coc21( bM[2],bM[4] );
if IsZero( a ) and not IsZero( c ) then # make a not zero
bM[1]:= bM[1]+bM[2];
a:= coc21( bM[1],bM[3] );
b:= coc21( bM[1],bM[4] );
fi;
if not IsZero( a ) then
bM[1]:= bM[1]/a;
bM[2]:= bM[2]*a;
c:= coc21( bM[2],bM[3] );
bM[2]:= bM[2]-c*bM[1];
b:= coc21( bM[1],bM[4] );
d:= coc21( bM[2],bM[4] );
if not IsZero( d ) then
bM[4]:= bM[4]/d-b/d*bM[3];
else
bM[4]:= bM[4]-b*bM[3];
fi;
else # here also c= 0.
m:= LieAlgDBHelper.comp_mat( b, d, One( F ) );
store:= bM[1];
bM[1]:= m[1][1]*bM[1]+m[1][2]*bM[2];
bM[2]:= m[2][1]*store+m[2][2]*bM[2];
fi;
s13:= coc21( bM[1],bM[3] );
s24:= coc21( bM[2],bM[4] );
s14:= coc21( bM[1],bM[4] );
r:= 1; t:= 0;
u:= 0; v:= 1;
if not IsZero( s13 ) then
if not IsZero( s24 ) then
a:= coc22( bM[1],bM[3] );
u:= -a; v:= 1;
f1:= coc22;
f5:= coc21;
coc22:= function( u,v ) return -a*f5( u,v )+f1( u,v ); end;
b:= coc22( bM[1],bM[4] );
c:= coc22( bM[2],bM[3] );
d:= coc22( bM[2],bM[4] );
if IsZero( c ) then
if not IsZero( d ) then
bM[3]:= bM[3]/d;
bM[2]:= bM[2]/d;
bM[1]:= bM[1]-b*bM[2];
bM[4]:= bM[4]+b*bM[3];
cf:= 1/coc21( bM[1],bM[3] );
f2:= coc21;
coc21:= function( u,v ) return cf*f2( u,v ); end;
r:= r*cf; t:= t*cf;
cf1:= 1/coc22( bM[2],bM[4] );
f3:= coc22;
coc22:= function( u,v ) return cf1*f3( u,v ); end;
u:= u*cf1; v:= v*cf1;
f4:= coc21;
coc21:= function( u,v )
return f4( u,v )-coc22( u,v );
end;
r:= r-u; t:= t-v;
c1:= coc21( bM[1],bM[2] );
c2:= coc22( bM[1],bM[2] );
T:= EmptySCTable( 6, Zero( F ), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,5] );
SetEntrySCTable( T, 2, 4, [1,6] );
N:= LieAlgebraByStructureConstants( F, T );
bM:= Basis( M, bM );
imgs:= [ ];
for x in Basis( K ) do
cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) );
cz1:= ShallowCopy( cz );
cz1[1]:= r*cz[1]+t*cz[2];
cz1[2]:= u*cz[1]+v*cz[2];
cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p,x ) ) ) );
Append( cx, cz1 );
# subtract the cocycle
cx[5]:= cx[5]-c1*cx[3];
cx[6]:= cx[6]-c2*cx[3];
Add( imgs, LinearCombination( Basis( N ), cx ) );
od;
return rec( type:= [ 6, 27 ], f:= LieAlgDBHelper.liealg_hom( K,N,Basis( K ),imgs ) );
else # d= 0
cf:= 1/coc21( bM[1],bM[3] );
f2:= coc21;
coc21:= function( u,v ) return cf*f2( u,v ); end;
r:= r*cf; t:= t*cf;
cf1:= 1/coc22( bM[1],bM[4] );
f3:= coc22;
coc22:= function( u,v ) return cf1*f3( u,v ); end;
u:= u*cf1; v:= v*cf1;
c1:= coc21( bM[1],bM[2] );
c2:= coc22( bM[1],bM[2] );
T:= EmptySCTable( 6, Zero( F ), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,5] );
SetEntrySCTable( T, 1, 4, [1,6] );
SetEntrySCTable( T, 2, 4, [1,5] );
N:= LieAlgebraByStructureConstants( F, T );
bM:= Basis( M, bM );
imgs:= [ ];
for x in Basis( K ) do
cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) );
cz1:= ShallowCopy( cz );
cz1[1]:= r*cz[1]+t*cz[2];
cz1[2]:= u*cz[1]+v*cz[2];
cx:= ShallowCopy( Coefficients( bM, Image( tau,Image( p,x ) ) ) );
Append( cx, cz1 );
# subtract the cocycle
cx[5]:= cx[5]-c1*cx[3];
cx[6]:= cx[6]-c2*cx[3];
Add( imgs, LinearCombination( Basis( N ), cx ) );
od;
return rec( type:= [ 6, 23 ], f:= LieAlgDBHelper.liealg_hom( K,N,Basis( K ),imgs ) );
fi;
else # i.e., c not 0
if IsZero( d ) then
bM[1]:= bM[1]/c; bM[3]:= bM[3]/c; bM[4]:= bM[4]/( c^2 );
cf:= 1/coc21( bM[1],bM[3] );
f2:= coc21;
coc21:= function( u,v ) return cf*f2( u,v ); end;
r:= r*cf; t:= t*cf;
b:= coc22( bM[1],bM[4] );
c1:= coc21( bM[1],bM[2] );
c2:= coc22( bM[1],bM[2] );
bb:= ShallowCopy( b );
if not IsZero( b ) then
if IsFinite( F ) then
if not Characteristic( F )= 2 then
xi:= PrimitiveElement( F );
ex:= LogFFE( b, xi );
if IsEvenInt( ex ) then
b:= One( F );
eps:= xi^( ex/2 );
else
b:= xi;
eps:= xi^( ( ex-1 )/2 );
fi;
else
eps:= b^( Size( F )/2 );
b:= Zero( F );
fi;
elif F = Rationals then
uu:= LieAlgDBHelper.sq_free( b );
eps:= uu[1];
b:= uu[2];
else
eps:= 1;
fi;
fi;
T:= EmptySCTable( 6, Zero( F ), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,5] );
SetEntrySCTable( T, 1, 4, [b,6] );
SetEntrySCTable( T, 2, 3, [1,6] );
SetEntrySCTable( T, 2, 4, [1,5] );
N:= LieAlgebraByStructureConstants( F, T );
bM:= Basis( M, bM );
imgs:= [ ];
for x in Basis( K ) do
cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) );
cz1:= ShallowCopy( cz );
cz1[1]:= r*cz[1]+t*cz[2];
cz1[2]:= u*cz[1]+v*cz[2];
cx:= ShallowCopy( Coefficients( bM, Image( tau,Image( p,x ) ) ) );
Append( cx, cz1 );
# subtract the cocycle
cx[5]:= cx[5]-c1*cx[3];
cx[6]:= cx[6]-c2*cx[3];
if not IsZero( bb ) then
# isom
cx[1]:= cx[1]*eps;
cx[3]:= cx[3]*eps;
cx[4]:= cx[4]*eps^2;
cx[5]:= cx[5]*eps^2;
cx[6]:= cx[6]*eps;
if Characteristic( F )= 2 then
# isom
cx1:= ShallowCopy( cx );
cx1[1]:= cx[1];
cx1[2]:= cx[1]+cx[2];
cx1[3]:= cx[3]+cx[4];
cx1[4]:= cx[4];
cx1[5]:= cx[5];
cx1[6]:= cx[5]+cx[6];
cx:= ShallowCopy( cx1 );
fi;
fi;
Add( imgs, LinearCombination( Basis( N ), cx ) );
od;
return rec( type:= [ 6, 24, b ], f:= LieAlgDBHelper.liealg_hom( K,N,Basis( K ),imgs ) );
else # i. e., d not 0
if not Characteristic( F )= 2 then
bM[1]:= ( 2*c/d )*bM[1]+bM[2];
bM[2]:= d/( 2*c )*bM[2];
bM[4]:= ( 2*c/d )^2*bM[4]-( 2*c/d )*bM[3];
cf:= 1/coc21( bM[1],bM[3] );
f2:= coc21;
coc21:= function( u,v ) return cf*f2( u,v ); end;
r:= r*cf; t:= t*cf;
cf1:= coc22( bM[2],bM[3] );
f3:= coc22;
coc22:= function( u,v ) return ( f3( u,v )-c*coc21( u,v ) )/cf1; end;
u:= ( u-c*r )/cf1; v:= ( v-c*t )/cf1;
b:= coc22( bM[1],bM[4] );
c1:= coc21( bM[1],bM[2] );
c2:= coc22( bM[1],bM[2] );
bb:= ShallowCopy( b );
if not IsZero( b ) then
if IsFinite( F ) then
xi:= PrimitiveElement( F );
ex:= LogFFE( b, xi );
if IsEvenInt( ex ) then
b:= One( F );
eps:= xi^( ex/2 );
else
b:= xi;
eps:= xi^( ( ex-1 )/2 );
fi;
elif F = Rationals then
uu:= LieAlgDBHelper.sq_free( b );
eps:= uu[1];
b:= uu[2];
else
eps:= 1;
fi;
fi;
T:= EmptySCTable( 6, Zero( F ), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,5] );
SetEntrySCTable( T, 1, 4, [b,6] );
SetEntrySCTable( T, 2, 3, [1,6] );
SetEntrySCTable( T, 2, 4, [1,5] );
N:= LieAlgebraByStructureConstants( F, T );
bM:= Basis( M, bM );
imgs:= [ ];
for x in Basis( K ) do
cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) );
cz1:= ShallowCopy( cz );
cz1[1]:= r*cz[1]+t*cz[2];
cz1[2]:= u*cz[1]+v*cz[2];
cx:= ShallowCopy( Coefficients( bM, Image( tau,Image( p,x ) ) ) );
Append( cx, cz1 );
# subtract the cocycle
cx[5]:= cx[5]-c1*cx[3];
cx[6]:= cx[6]-c2*cx[3];
if not IsZero( bb ) then
# isom
cx[1]:= cx[1]*eps;
cx[3]:= cx[3]*eps;
cx[4]:= cx[4]*eps^2;
cx[5]:= cx[5]*eps^2;
cx[6]:= cx[6]*eps;
fi;
Add( imgs, LinearCombination( Basis( N ), cx ) );
od;
return rec( type:= [ 6, 24, b ], f:= LieAlgDBHelper.liealg_hom( K,N,Basis( K ),imgs ) );
else
bM[1]:= c*bM[1];
bM[2]:= d*bM[2];
bM[3]:= c*d*bM[3];
bM[4]:= c^2*bM[4];
cf:= 1/coc21( bM[1],bM[3] );
f2:= coc21;
coc21:= function( u,v ) return cf*f2( u,v ); end;
r:= r*cf; t:= t*cf;
cf1:= coc22( bM[2],bM[3] );
f3:= coc22;
coc22:= function( u,v ) return f3( u,v )/cf1; end;
u:= u/cf1; v:= v/cf1;
b:= coc22( bM[1],bM[4] );
c1:= coc21( bM[1],bM[2] );
c2:= coc22( bM[1],bM[2] );
bb:= ShallowCopy( b );
if not IsZero( b ) then
if IsFinite( F ) then
ex:= First( [0..( Size( F )-1 )],i->Trace( F,PrimitiveElement( F )^i )= One( F ) );
xi:= PrimitiveElement( F )^ex;
P:= PolynomialRing( F,["y"] );
y:= P.1;
if Trace( F,b )= Zero( F ) then
root:= Factors( P,y^2+y+b )[1];
eps:= CoefficientsOfUnivariatePolynomial( root )[1];
b:= Zero( F );
else
root:= Factors( P,y^2+y+b+xi )[1];
eps:= CoefficientsOfUnivariatePolynomial( root )[1];
b:= xi;
fi;
else
eps:= 1;
fi;
fi;
T:= EmptySCTable( 6, Zero( F ), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,5] );
SetEntrySCTable( T, 1, 4, [b,6] );
SetEntrySCTable( T, 2, 3, [1,6] );
SetEntrySCTable( T, 2, 4, [1,5,1,6] );
N:= LieAlgebraByStructureConstants( F, T );
bM:= Basis( M, bM );
imgs:= [ ];
for x in Basis( K ) do
cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) );
cz1:= ShallowCopy( cz );
cz1[1]:= r*cz[1]+t*cz[2];
cz1[2]:= u*cz[1]+v*cz[2];
cx:= ShallowCopy( Coefficients( bM, Image( tau,Image( p,x ) ) ) );
Append( cx, cz1 );
# subtract the cocycle
cx[5]:= cx[5]-c1*cx[3];
cx[6]:= cx[6]-c2*cx[3];
if not IsZero( bb ) then
# isom
cx1:= ShallowCopy( cx );
cx1[1]:= cx[1];
cx1[2]:= +cx[1]*eps+cx[2];
cx1[3]:= cx[3]+cx[4]*eps;
cx1[4]:= cx[4];
cx1[5]:= cx[5];
cx1[6]:= cx[5]*eps+cx[6];
cx:= ShallowCopy( cx1 );
fi;
Add( imgs, LinearCombination( Basis( N ), cx ) );
od;
return rec( type:= [ 6, 36, b ], f:= LieAlgDBHelper.liealg_hom( K,N,Basis( K ),imgs ) );
fi;
fi;
fi;
else # theta_1 = Delta_13
b:= coc22( bM[1],bM[4] );
d:= coc22( bM[2],bM[4] );
if not IsZero( d ) then
bM[1]:= bM[1]-( b/d )*bM[2];
bM[4]:= bM[4]/d;
c:= coc22( bM[2],bM[3] );
if not IsZero( c ) then
bM[1]:= bM[1]/c;
bM[3]:= bM[3]/c;
# normalise coc21
cf:= 1/coc21( bM[1],bM[3] );
f1:= coc21;
coc21:= function( u,v ) return cf*f1( u,v ); end;
r:= r*cf; t:= t*cf;
#make a = 0 by subtracting...
a:= coc22( bM[1],bM[3] );
u:= u-a*r; v:= v-a*t;
f2:= coc22;
coc22:= function( u,v ) return -a*coc21( u,v )+f2( u,v ); end;
c1:= coc21( bM[1],bM[2] );
c2:= coc22( bM[1],bM[2] );
if not Characteristic( F )= 2 then
T:= EmptySCTable( 6, Zero( F ), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,5] );
SetEntrySCTable( T, 1, 4, [1,6] );
SetEntrySCTable( T, 2, 3, [1,6] );
SetEntrySCTable( T, 2, 4, [1,5] );
N:= LieAlgebraByStructureConstants( F, T );
bM:= Basis( M, bM );
imgs:= [ ];
for x in Basis( K ) do
cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) );
cz1:= ShallowCopy( cz );
cz1[1]:= r*cz[1]+t*cz[2];
cz1[2]:= u*cz[1]+v*cz[2];
cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) );
Append( cx, cz1 );
# subtract the cocycle
cx[5]:= cx[5]-c1*cx[3];
cx[6]:= cx[6]-c2*cx[3];
#isom...
cx1:= ShallowCopy( cx );
cx1[1]:= cx[1]+cx[2];
cx1[2]:= -cx[1] +cx[2];
cx1[3]:= 2*cx[3]+cx[4];
cx1[4]:= cx[4];
cx1[5]:= 2*cx[5]+2*cx[6];
cx1[6]:= -2*cx[5]+2*cx[6];
Add( imgs, LinearCombination( Basis( N ), cx1 ) );
od;
return rec( type:= [ 6, 24, One( F ) ], f:= LieAlgDBHelper.liealg_hom( K,N,Basis( K ),imgs ) );
else
T:= EmptySCTable( 6, Zero( F ), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,5] );
SetEntrySCTable( T, 2, 3, [1,6] );
SetEntrySCTable( T, 2, 4, [1,5,1,6] );
N:= LieAlgebraByStructureConstants( F, T );
bM:= Basis( M, bM );
imgs:= [ ];
for x in Basis( K ) do
cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) );
cz1:= ShallowCopy( cz );
cz1[1]:= r*cz[1]+t*cz[2];
cz1[2]:= u*cz[1]+v*cz[2];
cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) );
Append( cx, cz1 );
# subtract the cocycle
cx[5]:= cx[5]-c1*cx[3];
cx[6]:= cx[6]-c2*cx[3];
#isom...
cx1:= ShallowCopy( cx );
cx1[1]:= cx[1]+cx[2];
cx1[2]:= cx[1];
cx1[3]:= cx[3]+cx[4];
cx1[4]:= cx[4];
cx1[5]:= cx[5]+cx[6];
cx1[6]:= cx[5];
Add( imgs, LinearCombination( Basis( N ), cx1 ) );
od;
return rec( type:= [ 6, 36, Zero( F ) ], f:= LieAlgDBHelper.liealg_hom( K,N,Basis( K ),imgs ) );
fi;
else # c= 0
# normalise coc21
cf:= 1/coc21( bM[1],bM[3] );
f1:= coc21;
coc21:= function( u,v ) return cf*f1( u,v ); end;
r:= r*cf; t:= t*cf;
#make a = 0 by subtracting...
a:= coc22( bM[1],bM[3] );
u:= u-a*r; v:= v-a*t;
f2:= coc22;
coc22:= function( u,v ) return -a*coc21( u,v )+f2( u,v ); end;
c1:= coc21( bM[1],bM[2] );
c2:= coc22( bM[1],bM[2] );
T:= EmptySCTable( 6, Zero( F ), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,5] );
SetEntrySCTable( T, 2, 4, [1,6] );
N:= LieAlgebraByStructureConstants( F, T );
bM:= Basis( M, bM );
imgs:= [ ];
for x in Basis( K ) do
cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) );
cz1:= ShallowCopy( cz );
cz1[1]:= r*cz[1]+t*cz[2];
cz1[2]:= u*cz[1]+v*cz[2];
cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) );
Append( cx, cz1 );
# subtract the cocycle
cx[5]:= cx[5]-c1*cx[3];
cx[6]:= cx[6]-c2*cx[3];
Add( imgs, LinearCombination( Basis( N ), cx ) );
od;
return rec( type:= [ 6, 27 ], f:= LieAlgDBHelper.liealg_hom( K,N,Basis( K ),imgs ) );
fi;
else # d= 0
bM[4]:= bM[4]/b; # now b= 1
c:= coc22( bM[2],bM[3] );
if not IsZero( c ) then
bM[1]:= bM[1]/c; bM[3]:= bM[3]/c; bM[4]:= c*bM[4];
# normalise coc21
cf:= 1/coc21( bM[1],bM[3] );
f1:= coc21;
coc21:= function( u,v ) return cf*f1( u,v ); end;
r:= r*cf; t:= t*cf;
#make a = 0 by subtracting...
a:= coc22( bM[1],bM[3] );
u:= u-a*r; v:= v-a*t;
f2:= coc22;
coc22:= function( u,v ) return -a*coc21( u,v )+f2( u,v ); end;
c1:= coc21( bM[1],bM[2] );
c2:= coc22( bM[1],bM[2] );
T:= EmptySCTable( 6, Zero( F ), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,5] );
SetEntrySCTable( T, 2, 3, [1,6] );
SetEntrySCTable( T, 2, 4, [1,5] );
N:= LieAlgebraByStructureConstants( F, T );
bM:= Basis( M, bM );
imgs:= [ ];
for x in Basis( K ) do
cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) );
cz1:= ShallowCopy( cz );
cz1[1]:= r*cz[1]+t*cz[2];
cz1[2]:= u*cz[1]+v*cz[2];
cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) );
Append( cx, cz1 );
# subtract the cocycle
cx[5]:= cx[5]-c1*cx[3];
cx[6]:= cx[6]-c2*cx[3];
#isom...
cx1:= ShallowCopy( cx );
cx1[1]:= -cx[2];
cx1[2]:= -cx[1];
cx1[3]:= -cx[3];
cx1[4]:= -cx[4];
cx1[5]:= cx[6];
cx1[6]:= cx[5];
Add( imgs, LinearCombination( Basis( N ), cx1 ) );
od;
return rec( type:= [ 6, 24, Zero( F ) ], f:= LieAlgDBHelper.liealg_hom( K,N,Basis( K ),imgs ) );
else # c= 0
# normalise coc21
cf:= 1/coc21( bM[1],bM[3] );
f1:= coc21;
coc21:= function( u,v ) return cf*f1( u,v ); end;
r:= r*cf; t:= t*cf;
#make a = 0 by subtracting...
a:= coc22( bM[1],bM[3] );
u:= u-a*r; v:= v-a*t;
f2:= coc22;
coc22:= function( u,v ) return -a*coc21( u,v )+f2( u,v ); end;
c1:= coc21( bM[1],bM[2] );
c2:= coc22( bM[1],bM[2] );
T:= EmptySCTable( 6, Zero( F ), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,5] );
SetEntrySCTable( T, 1, 4, [1,6] );
N:= LieAlgebraByStructureConstants( F, T );
bM:= Basis( M, bM );
imgs:= [ ];
for x in Basis( K ) do
cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) );
cz1:= ShallowCopy( cz );
cz1[1]:= r*cz[1]+t*cz[2];
cz1[2]:= u*cz[1]+v*cz[2];
cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) );
Append( cx, cz1 );
# subtract the cocycle
cx[5]:= cx[5]-c1*cx[3];
cx[6]:= cx[6]-c2*cx[3];
Add( imgs, LinearCombination( Basis( N ), cx ) );
od;
return rec( type:= [ 6, 25 ], f:= LieAlgDBHelper.liealg_hom( K,N,Basis( K ),imgs ) );
fi;
fi;
fi;
else # i.e., theta_1 = Delta_14
c:= coc22( bM[2],bM[3] );
if not IsZero( c ) then
bM[1]:= bM[1]/c; bM[3]:= bM[3]/c;
a:= coc22( bM[1],bM[3] );
d:= coc22( bM[2],bM[4] );
bM[1]:= bM[1]-a*bM[2];
bM[4]:= bM[4]-d*bM[3];
# normalise coc21
cf:= 1/coc21( bM[1],bM[4] );
f1:= coc21;
coc21:= function( u,v ) return cf*f1( u,v ); end;
r:= r*cf; t:= t*cf;
#make b = 0 by subtracting...
b:= coc22( bM[1],bM[4] );
u:= u-b*r; v:= v-b*t;
f2:= coc22;
coc22:= function( u,v ) return -b*coc21( u,v )+f2( u,v ); end;
c1:= coc21( bM[1],bM[2] );
c2:= coc22( bM[1],bM[2] );
T:= EmptySCTable( 6, Zero( F ), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,5] );
SetEntrySCTable( T, 2, 4, [1,6] );
N:= LieAlgebraByStructureConstants( F, T );
bM:= Basis( M, bM );
imgs:= [ ];
for x in Basis( K ) do
cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) );
cz1:= ShallowCopy( cz );
cz1[1]:= r*cz[1]+t*cz[2];
cz1[2]:= u*cz[1]+v*cz[2];
cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) );
Append( cx, cz1 );
# subtract the cocycle
cx[5]:= cx[5]-c1*cx[3];
cx[6]:= cx[6]-c2*cx[3];
#isom...
cx1:= ShallowCopy( cx );
cx1[1]:= -cx[2];
cx1[2]:= cx[1];
cx1[3]:= cx[3];
cx1[4]:= -cx[4];
cx1[5]:= -cx[6];
cx1[6]:= -cx[5];
Add( imgs, LinearCombination( Basis( N ), cx1 ) );
od;
return rec( type:= [ 6, 27 ], f:= LieAlgDBHelper.liealg_hom( K,N,Basis( K ),imgs ) );
else #c= 0
a:= coc22( bM[1],bM[3] );
bM[2]:= bM[2]/a; bM[3]:= bM[3]/a;
d:= coc22( bM[2],bM[4] );
if not IsZero( d ) then
bM[4]:= bM[4]/d;
fi;
# normalise coc21
cf:= 1/coc21( bM[1],bM[4] );
f1:= coc21;
coc21:= function( u,v ) return cf*f1( u,v ); end;
r:= r*cf; t:= t*cf;
#make b = 0 by subtracting...
b:= coc22( bM[1],bM[4] );
u:= u-b*r; v:= v-b*t;
f2:= coc22;
coc22:= function( u,v ) return -b*coc21( u,v )+f2( u,v ); end;
c1:= coc21( bM[1],bM[2] );
c2:= coc22( bM[1],bM[2] );
T:= EmptySCTable( 6, Zero( F ), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,5] );
SetEntrySCTable( T, 1, 4, [1,6] );
if not IsZero( d ) then
SetEntrySCTable( T, 2, 4, [1,5] );
fi;
N:= LieAlgebraByStructureConstants( F, T );
bM:= Basis( M, bM );
imgs:= [ ];
for x in Basis( K ) do
cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) );
cz1:= ShallowCopy( cz );
cz1[1]:= r*cz[1]+t*cz[2];
cz1[2]:= u*cz[1]+v*cz[2];
cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) );
Append( cx, cz1 );
# subtract the cocycle
cx[5]:= cx[5]-c1*cx[3];
cx[6]:= cx[6]-c2*cx[3];
#isom...
cx1:= ShallowCopy( cx );
cx1[5]:= cx[6];
cx1[6]:= cx[5];
Add( imgs, LinearCombination( Basis( N ), cx1 ) );
od;
if not IsZero( d ) then
return rec( type:= [ 6, 23 ], f:= LieAlgDBHelper.liealg_hom( K,N,Basis( K ),imgs ) );
else
return rec( type:= [ 6, 25 ], f:= LieAlgDBHelper.liealg_hom( K,N,Basis( K ),imgs ) );
fi;
fi;
fi;
elif type = [ 4, 3 ] then
coc21:= function( u, v ) return Coefficients( Basis( C ), coc2( u,v ) )[1]; end;
coc22:= function( u, v ) return Coefficients( Basis( C ), coc2( u,v ) )[2]; end;
bM:= ShallowCopy( BasisVectors( Basis( M ) ) );
a:= coc21( bM[1],bM[4] );
b:= coc21( bM[2],bM[3] );
c:= coc22( bM[1],bM[4] );
d:= coc22( bM[2],bM[3] );
m:= [[a,b],[c,d]]^-1;
r:= m[1][1]; t:= m[1][2];
u:= m[2][1]; v:= m[2][2];
cc:= coc21;
f1:= coc22;
coc21:= function( u,v ) return r*cc( u,v )+t*f1( u,v ); end;
coc22:= function( x,y ) return u*cc( x,y )+v*f1( x,y ); end;
c1:= coc21( bM[1],bM[2] );
c2:= coc22( bM[1],bM[2] );
c3:= coc21( bM[1],bM[3] );
c4:= coc22( bM[1],bM[3] );
T:= EmptySCTable( 6, Zero( F ), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,3] );
SetEntrySCTable( T, 1, 3, [1,4] );
SetEntrySCTable( T, 1, 4, [1,5] );
SetEntrySCTable( T, 2, 3, [1,6] );
N:= LieAlgebraByStructureConstants( F, T );
bM:= Basis( M, bM );
imgs:= [ ];
for x in Basis( K ) do
cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) );
cz1:= ShallowCopy( cz );
cz1[1]:= r*cz[1]+t*cz[2];
cz1[2]:= u*cz[1]+v*cz[2];
cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) );
Append( cx, cz1 );
# subtract the cocycle
cx[5]:= cx[5]-c1*cx[3]-c3*cx[4];
cx[6]:= cx[6]-c2*cx[3]-c4*cx[4];
Add( imgs, LinearCombination( Basis( N ), cx ) );
od;
return rec( type:= [ 6, 28 ], f:= LieAlgDBHelper.liealg_hom( K,N,Basis( K ),imgs ) );
elif type = [ 3, 1 ] then
coc21:= function( u, v ) return Coefficients( Basis( C ), coc2( u,v ) )[1]; end;
coc22:= function( u, v ) return Coefficients( Basis( C ), coc2( u,v ) )[2]; end;
coc23:= function( u, v ) return Coefficients( Basis( C ), coc2( u,v ) )[3]; end;
bM:= Basis( M );
m:= [ [], [], [] ];
m[1][1]:= coc21( bM[1],bM[2] );
m[1][2]:= coc21( bM[1],bM[3] );
m[1][3]:= coc21( bM[2],bM[3] );
m[2][1]:= coc22( bM[1],bM[2] );
m[2][2]:= coc22( bM[1],bM[3] );
m[2][3]:= coc22( bM[2],bM[3] );
m[3][1]:= coc23( bM[1],bM[2] );
m[3][2]:= coc23( bM[1],bM[3] );
m[3][3]:= coc23( bM[2],bM[3] );
m:= m^-1;
T:= EmptySCTable( 6, Zero( F ), "antisymmetric" );
SetEntrySCTable( T, 1, 2, [1,4] );
SetEntrySCTable( T, 1, 3, [1,5] );
SetEntrySCTable( T, 2, 3, [1,6] );
N:= LieAlgebraByStructureConstants( F, T );
imgs:= [ ];
for x in Basis( K ) do
cz:= Coefficients( Basis( C ), x-s( Image( p,x ) ) );
cz:= cz*TransposedMat( m );
cx:= ShallowCopy( Coefficients( bM, Image( tau, Image( p, x ) ) ) );
Append( cx, cz );
Add( imgs, LinearCombination( Basis( N ), cx ) );
od;
return rec( type:= [ 6, 26 ], f:= LieAlgDBHelper.liealg_hom( K,N,Basis( K ),imgs ) );
fi;
end;
LieAlgDBHelper.nilp_type:= function( L )
local r, t, a, g, exp, c, K, imgs, pp, s, f, F, name;
if (not Dimension(L) in [5,6]) or (not IsLieNilpotent(L)) then
Error("This function is only implemented for nilpotent Lie algebras of dim 5,6");
fi;
if Dimension(L) = 5 then
r:= LieAlgDBHelper.class_dim_5( L );
else
r:= LieAlgDBHelper.class_dim_6( L );
fi;
F:= LeftActingDomain(L);
name:= "N";
Append( name, String(r.type[1]) );
Append( name, "_" );
Append( name, String( r.type[2] ) );
Append( name, "( ");
Append( name, LieAlgDBField2String( F ) );
if Length(r.type) = 3 then
Append( name, ", " );
Append( name, String( r.type[3] ) );
pp:= [r.type[3]];
else
pp:= [ ];
fi;
Append( name, " )" );
return rec( name:= [ name, pp ], isom:= r.f );
end;
InstallMethod( LieAlgebraIdentification, "for a Lie algebra",
true,
[ IsLieAlgebra ], 0,
function( L )
local n, r;
n:= Dimension(L);
if n in [2,3,4] then
if not IsLieSolvable(L) then Error("<L> has to be a solvable Lie algebra"); fi;
r:= LieAlgDBHelper.solv_type(L);
elif n in [5,6] then
if not IsLieNilpotent(L) then Error("<L> has to be a nilpotent Lie algebra"); fi;
r:= LieAlgDBHelper.nilp_type(L);
else
Error("the dimension has to satisfy 2<= dim <= 6");
fi;
return rec( name:= r.name[1], parameters:= r.name[2], isomorphism:= r.isom );
end );
InstallMethod( AllSolvableLieAlgebras,
"for a finite field and a positive integer",
true,
[ IsField and IsFinite, IsPosInt ], 0,
function( F, dim )
local parlist, R, fam;
if not IsFinite(F) then Error("F has to be a finite field"); fi;
if not dim in [1,2,3,4] then Error("solvable Lie algebras of this dimension are not included"); fi;
if dim = 1 then
parlist := [[1,1]];
elif dim = 2 then
parlist:= [[2,1],[2,2]];
elif dim = 3 then
parlist := EnumeratorByFunctions( NewFamily( IsList ),
rec(
ElementNumber := function( e, x )
local aa;
aa:= [ Zero(F), One(F) ];
if Characteristic( F ) > 2 then
Add( aa, PrimitiveRoot(F) );
fi;
if x in [1,2] then
return [3,x];
elif x in [3..Size( F )+2] then
return [3,3,Enumerator( F )[x-2]];
elif Characteristic( F ) = 2 and x in [Size( F )+3..Size( F )+4]
then return [3,4,aa[x-Size(F)-2]];
elif Characteristic( F ) <> 2 and x in [Size( F )+3..Size( F )+5]
then return [3,4,aa[x-Size(F)-2]];
fi;
end,
NumberElement := function( e, x )
local list1, list2;
list1 := [[3,1],[3,2]];
list2 := [[3,4,Zero(F)],[3,4,One(F)]];
if Characteristic( F ) > 2 then
Add( list2, [3,4,PrimitiveRoot(F)]);
fi;
if x in list1 then
return Position( list1, x );
elif x{[1,2]} = [3,3] then
return Position( Enumerator( F ), x[3] ) + 2;
elif x in list2 then
return Position( list2, x ) + Size( F )+2;
fi;
end,
Length := function( e )
if Characteristic( F ) = 2 then
return Size( F )+4;
else
return Size( F )+5;
fi;
end ));
elif dim = 4 then
parlist := EnumeratorByFunctions( NewFamily( IsList ),
rec(
ElementNumber := function( e, x )
local a, T, i, ff, q, l1, l2, l3, l4, l5;
q:= Size( F ); a := PrimitiveRoot( F );
l1 := [[4,7, Zero(F), One(F)],[4,7, Zero(F), a],[4,14,One(F)],
[4,14,a],[4,7,a^0, Zero(F)],[4,7, a, Zero(F)],
[4,7, a^2, Zero(F)]];
l2 := [[4,7, Zero(F), One(F)],[4,11, One(F), Zero(F) ],
[4,14, One(F)],[4,7, One(F), Zero(F)]];
l3 := [[4,7, Zero(F), One(F)],[4,7, Zero(F), a],[4,14,One(F)],
[4,14,a],[4,7, One(F), Zero(F)]];
l4 := [[4,7, Zero(F), One(F)],[4,11, One(F), Zero(F) ],
[4,14, One(F)],[4,7, a^0, Zero(F)],
[4,7, a, Zero(F)],[4,7, a^2, Zero(F)]];
l5 := [[4,7, Zero(F), One(F)],[4,7, Zero(F), a],[4,14,One(F)],
[4,14,a],[4,7, One(F), Zero(F)]];
if x in [1,2] then
return [4,x];
elif x in [3..Size( F )+2] then
return [4,3,Enumerator( F )[x-2]];
elif x in [Size( F )+3..Size( F )+4] then
return [4,x-Size(F)+1];
elif x in [Size(F)+5..Size(F)^2+Size(F)+4] then
return [4,6,
Enumerator( F )[Int((x-Size( F )-5)/Size( F ))+1],
Enumerator( F )[((x-Size(F)-5) mod Size( F ))+1]];
elif x in [Size(F)^2+Size(F)+5..Size(F)^2+2*Size(F)+4] then
return [4,7,Enumerator( F )[x-(Size(F)^2+Size(F)+4)],
Enumerator( F )[x-(Size(F)^2+Size(F)+4)]];
elif x = Size(F)^2+2*Size(F)+5 then
return [4,8];
elif x = Size(F)^2+2*Size(F)+6 then
a:= PrimitiveRoot( F );
T:= Indeterminate(F);
for i in [1..q-1] do
ff:= Factors( T^2-T-a^i );
if Length(ff) = 1 then
return [4,9,a^i];
break;
fi;
od;
elif x = Size(F)^2+2*Size(F)+7 then
return [4,12];
elif x in [Size(F)^2+2*Size(F)+8..Size(F)^2+3*Size(F)+7] then
return [4,13,Enumerator( F )[x-(Size(F)^2+2*Size(F)+7)]];
elif q mod 6 = 1 and
x in [Size(F)^2+3*Size(F)+8..Size(F)^2+3*Size(F)+14] then
return l1[x-(Size(F)^2+3*Size(F)+7)];
elif q mod 6 = 2 and
x in [Size(F)^2+3*Size(F)+8..Size(F)^2+3*Size(F)+11] then
return l2[x-(Size(F)^2+3*Size(F)+7)];
elif q mod 6 = 3 and
x in [Size(F)^2+3*Size(F)+8..Size(F)^2+3*Size(F)+12] then
return l3[x-(Size(F)^2+3*Size(F)+7)];
elif q mod 6 = 4 and
x in [Size(F)^2+3*Size(F)+8..Size(F)^2+3*Size(F)+13] then
return l4[x-(Size(F)^2+3*Size(F)+7)];
elif q mod 6 = 5 and
x in [Size(F)^2+3*Size(F)+8..Size(F)^2+3*Size(F)+12] then
return l5[x-(Size(F)^2+3*Size(F)+7)];
fi;
end,
NumberElement := function( e, x )
local q, a, l1, l2, l3, l4, l5;
q:= Size( F ); a := PrimitiveRoot( F );
l1 := [[4,7, Zero(F), One(F)],[4,7, Zero(F), a],[4,14,One(F)],
[4,14,a],[4,7,a^0, Zero(F)],[4,7, a, Zero(F)],
[4,7, a^2, Zero(F)]];
l2 := [[4,7, Zero(F), One(F)],[4,11, One(F), Zero(F) ],
[4,14, One(F)],[4,7, One(F), Zero(F)]];
l3 := [[4,7, Zero(F), One(F)],[4,7, Zero(F), a],[4,14,One(F)],
[4,14,a],[4,7, One(F), Zero(F)]];
l4 := [[4,7, Zero(F), One(F)],[4,11, One(F), Zero(F) ],
[4,14, One(F)],[4,7, a^0, Zero(F)],
[4,7, a, Zero(F)],[4,7, a^2, Zero(F)]];
l5 := [[4,7, Zero(F), One(F)],[4,7, Zero(F), a],[4,14,One(F)],
[4,14,a],[4,7, One(F), Zero(F)]];
if x[2] in [1,2] then
return x[2];
elif x[2] = 3 then
return Position( Enumerator( F ), x[3] )+2;
elif x[2] in [4,5] then
return Size( F )+x[2]-1;
elif x[2] = 6 then
return Size( F )+4+(Position( Enumerator( F ), x[3])-1)*
Size( F )+Position( Enumerator( F ), x[4] );
elif x[2] = 7 and x[3]=x[4] then
return Size(F)^2+Size(F)+4+Position( Enumerator( F ), x[3] );
elif x[2] = 8 then
return Size(F)^2+2*Size(F)+5;
elif x[2] = 9 then
return Size(F)^2+2*Size(F)+6;
elif x[2] = 12 then
return Size(F)^2+2*Size(F)+7;
elif x[2] = 13 then
return Size(F)^2+2*Size(F)+7+
Position( Enumerator( F ), x[3] );
elif q mod 6 = 1 and x in l1 then
return Size(F)^2+3*Size(F)+7+Position( l1, x );
elif q mod 6 = 2 and x in l2 then
return Size(F)^2+3*Size(F)+7+Position( l2, x );
elif q mod 6 = 3 and x in l3 then
return Size(F)^2+3*Size(F)+7+Position( l3, x );
elif q mod 6 = 4 and x in l4 then
return Size(F)^2+3*Size(F)+7+Position( l4, x );
elif q mod 6 = 5 and x in l5 then
return Size(F)^2+3*Size(F)+7+Position( l5, x );
fi;
end,
Length := function( e )
local q;
q := Size( F );
if q mod 6 = 1 then
return q^2+3*q+14;
elif q mod 6 = 2 then
return q^2+3*q+11;
elif q mod 6 = 3 then
return q^2+3*q+12;
elif q mod 6 = 4 then
return q^2+3*q+13;
elif q mod 6 = 5 then
return q^2+3*q+12;
fi;
end ));
else
Error( "not yet implemented" );
fi;
R := rec( field := F,
dim := dim,
type := "Solvable",
parlist := parlist );
fam := NewFamily( IsLieAlgDBCollection_Solvable );
R := Objectify( NewType( fam, IsLieAlgDBCollection_Solvable ), R );
return R;
end );
InstallMethod( Enumerator,
"method for LieAlgDBCollections",
[ IsLieAlgDBCollection_Solvable ],
function( R )
return EnumeratorByFunctions( NewFamily(
CategoryCollections( IsLieAlgebra )),
rec(
ElementNumber := function( e, n )
local par;
par := R!.parlist[n];
return SolvableLieAlgebra( R!.field, par );
end,
NumberElement := function( e, x )
return Position( R!.parlist, x!.arg );
end,
Length := function( x ) return Length( R!.parlist ); end ));
end );
