Quelle nilpotent.gi
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#############################################################################
##
#W nilpotent.gi Small nilpotent Lie algebras Csaba Schneider
##
#W This file contains some functions to access the classifications
#W of small-dimensional nilpotent Lie algebras over small fields.
##
InstallMethod( NrNilpotentLieAlgebras,
"for a finite field and a positive integer",
true,
[ IsField and IsFinite, IsPosInt ],
0,
function( F, dim )
if dim = 1 then
return 1;
elif dim = 2 then
return 1;
elif dim = 3 then
return 2;
elif dim = 4 then
return 3;
elif dim = 5 then
return 9;
elif dim = 6 and Characteristic( F ) > 2 then
return 34;
elif dim = 6 and Characteristic( F )= 2 then
return 36;
elif dim = 7 and Size( F ) = 2 then
return 202;
elif dim = 7 and Size( F ) = 3 then
return 199;
elif dim = 7 and Size( F ) = 5 then
return 211;
elif dim = 8 and Size( F ) = 2 then
return 1831;
elif dim = 9 and Size( F ) = 2 then
return 27073;
else
Error( "The number of nilpotent Lie algebras is not known for these parameters" );
fi;
end );
#############################################################################
##
#F AllNilpotentLieAlgebras( F, dim )
##
InstallMethod( AllNilpotentLieAlgebras,
"for a finite field and a positive integer",
true,
[ IsField and IsFinite, IsPosInt ],
0,
function( F, dim )
local parlist, no, l, ex, xi, p, R, fam;
if dim = 1 then
parlist := [[1,1]];
elif dim = 2 then
parlist := [[2,1]];
elif dim = 3 then
parlist := [[3,1],[3,2]];
elif dim = 4 then
parlist := [[4,1],[4,2],[4,3]];
elif dim = 5 then
parlist := List( [1..9], x-> [5,x] );
elif dim = 6 and Characteristic( F ) > 2 then
parlist:= [ ];
p:= PrimitiveRoot( F );
for no in [1..28] do
if no in [ 19, 21 ] then
Add( parlist, [6,no,One(F)] );
Add( parlist, [6,no,p] );
elif no in [ 22, 24 ] then
Add( parlist, [6,no,0] );
Add( parlist, [6,no,One(F)] );
Add( parlist, [6,no,p] );
else
Add( parlist, [6,no] );
fi;
od;
elif dim = 6 and Characteristic( F ) = 2 then
parlist:= [ ];
ex:= First( [0..( Size( F )-1 )],i->Trace( F,PrimitiveElement( F )^i )= One( F ) );
xi:= PrimitiveElement( F )^ex;
l:=[1..36];
SubtractSet( l, [ 31, 32 ] );
for no in l do
if no in [ 19, 21 ] then
Add( parlist, [6,no,One(F)] );
elif no in [ 22, 24 ] then
Add( parlist, [6,no,0] );
elif no in [ 35, 36 ] then
Add( parlist, [6,no,0] );
Add( parlist, [6,no,xi] );
else
Add( parlist, [6,no] );
fi;
od;
elif [Size( F ), dim ] = [ 2, 7 ] then
if Length(_liealgdb_nilpotent_d7f2) = 0 then
UnbindGlobal( "_liealgdb_nilpotent_d7f2" );
ReadPackage( "liealgdb", "gap/nilpotent/nilpotent_data72.gi" );
fi;
parlist := _liealgdb_nilpotent_d7f2;
elif [Size( F ), dim ] = [ 2, 8 ] then
if Length(_liealgdb_nilpotent_d8f2) = 0 then
UnbindGlobal( "_liealgdb_nilpotent_d8f2" );
ReadPackage( "liealgdb", "gap/nilpotent/nilpotent_data82.gi" );
fi;
parlist := _liealgdb_nilpotent_d8f2;
elif [Size( F ), dim ] = [ 2, 9 ] then
if Length(_liealgdb_nilpotent_d9f2) = 0 then
UnbindGlobal( "_liealgdb_nilpotent_d9f2" );
ReadPackage( "liealgdb", "gap/nilpotent/nilpotent_data92.gi" );
fi;
parlist := _liealgdb_nilpotent_d9f2;
elif [Size( F ), dim ] = [ 3, 7 ] then
if Length(_liealgdb_nilpotent_d7f3) = 0 then
UnbindGlobal( "_liealgdb_nilpotent_d7f3" );
ReadPackage( "liealgdb", "gap/nilpotent/nilpotent_data73.gi" );
fi;
parlist := _liealgdb_nilpotent_d7f3;
elif [Size( F ), dim ] = [ 5, 7 ] then
if Length(_liealgdb_nilpotent_d7f5) = 0 then
UnbindGlobal( "_liealgdb_nilpotent_d7f5" );
ReadPackage( "liealgdb", "gap/nilpotent/nilpotent_data75.gi" );
fi;
parlist := _liealgdb_nilpotent_d7f5;
else
Error( "The list of nilpotent Lie algebras is not available for these parameters." );
fi;
R := rec( field := F,
dim := dim,
type := "Nilpotent",
parlist := parlist );
fam := NewFamily( IsLieAlgDBCollection_Nilpotent );
R := Objectify( NewType( fam, IsLieAlgDBCollection_Nilpotent ), R );
return R;
end );
BindGlobal( "ReadStringToNilpotentLieAlgebra", function( string, p, dim )
local digits, d, sum, i, coeffs, no_coeffs, T, pos, a, b, scentry, r, q, L;
digits := ['0','1','2','3','4','5','6','7','8','9',
'a','b','c','d','e','f','g','h','i','j',
'k','l','m','n','o','p','q','r','s','t',
'u','v','w','x','y','z','A','B','C','D',
'E','F','G','H','I','J','K','L','M','N',
'O','P','Q','R','S','T','U','V','W','X',
'Y','Z'];
d := 62^(Length( string ) - 1 );
sum := 0;
for i in string do
sum := sum + (Position( digits, i )-1)*d;
d := d/62;
od;
#Print( "number is ", sum, "\n" );
d := 1;
while d*p <= sum do
d := d*p;
od;
coeffs := [];
repeat
q := QuoInt( sum, d );
r := sum - q*d;
Add( coeffs, q );
sum := sum - q*d;
d := d/p;
until d = 1/p;
no_coeffs := Sum( Combinations( [1..dim], 2 ), x->(dim-x[2]));
coeffs := Concatenation( List( [1..no_coeffs-Length( coeffs )],
x->0 ), coeffs );
#Print( "Coeff list is: ", coeffs, "\n" );
T := EmptySCTable( dim, Zero( GF( p )), "antisymmetric" );
pos := 1;
for a in Reversed([1..dim-1]) do
for b in Reversed([a+1..dim]) do
scentry := [];
for i in Reversed( [b+1..dim] ) do
Append( scentry, [One(GF(p))*coeffs[pos], i] );
pos := pos + 1;
od;
SetEntrySCTable( T, b, a, scentry );
#Print( b, " ", a, ": ", scentry, "\n" );
od;
od;
L := LieAlgebraByStructureConstants( GF(p), T );
L!.arg := string;
Setter( IsLieNilpotent )( L, true );
return L;
end );
InstallMethod( Enumerator,
"method for LieAlgDBCollections",
[ IsLieAlgDBCollection_Nilpotent ],
function( R )
return EnumeratorByFunctions( NewFamily(
CategoryCollections( IsLieAlgebra )),
rec(
ElementNumber := function( e, n )
local par;
par := R!.parlist[n];
if IsString( par ) then
return ReadStringToNilpotentLieAlgebra( par, Size( R!.field ),
R!.dim );
else
return NilpotentLieAlgebra( R!.field, par );
fi;
end,
NumberElement := function( e, x )
return Position( R!.parlist, x!.arg );
end,
Length := function( x ) return Length( R!.parlist ); end ));
end );
[ Dauer der Verarbeitung: 0.66 Sekunden
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2026-03-28
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