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#############################################################################
##
#W lienp.gi Sophus package Csaba Schneider
##
#W Computing a nilpotent basis for a Lie algebra.
##
######################################################################
##
#P IsLieAlgebraWithNB( <L> )
##
## returns true if <L> has a nilpotent basis.
InstallMethod(
IsLieAlgebraWithNB,
"for nilpotent Lie algebras",
[ IsLieNilpotentOverFp ],
function( L )
if IsBound( L!.NilpotentBasis ) then
return true;
else
return false;
fi;
end );
######################################################################
##
#A NilpotentBasis( <L> )
##
## The function computes a nilpotent basis for a nilpotent Lie algebra.
## See the manual for explanation
InstallMethod(
NilpotentBasis,
"for nilpotent Lie algebras",
[ IsLieAlgebra ],
function( L )
local
i, j, k, # for indexing
f, # natural homom onto a quot of lcs
Q, # the image of f
newbas, # the new basis
defs, # definition of the new basis elements
weight, # the weight vector
choosebasis, # a function to choose a basis from a spanning set
firstlength, # number of weigtht 1 generators
lastlength, # the no. of generators with the last computed weight
indices, # indices of basis elements in a spanning set
class, # nilpotency class of L
list, # list to store coeffs
npb, # the new basis
S; # lower central series
choosebasis := function( S, s ) # S is a vectorspace s is a spanning set
local i, bas, S0, indices;
bas := []; indices := [];
S0 := Subspace( S, [] );
for i in [1..Length( s )] do
if not s[i] in S0 then
Add( bas, s[i] );
Add( indices, i );
S0 := Subspace( S, bas );
fi;
od;
return indices;
end;
if not IsLieNilpotentOverFp( L ) then
TryNextMethod();
fi;
S := LieLowerCentralSeries( L );
class := Length( S ) - 1;
f := NaturalHomomorphismByIdeal( L, S[2] );
Q := Image( f, L );
newbas := List( BasisVectors( Basis( Q )),
x->PreImagesRepresentative( f, x ));
defs := List( [1..Dimension( Q )], x -> 0 );
weight := List( [1..Dimension( Q )], x -> 1 );
firstlength := Dimension( Q ); lastlength := Dimension( Q );
for i in [2..class] do
f := NaturalHomomorphismByIdeal( S[i], S[i+1] );
Q := Image( f, S[i] );
list := [];
for j in [1..firstlength] do
for k in [ Maximum( j+1, Length( newbas ) - lastlength + 1 )..
Length( newbas )] do
Add( list, [ [j, k], newbas[j]*newbas[k]]);
od;
od;
indices := choosebasis( Q, List( list, x->Image( f, x[2] )));
for k in indices do
Add( newbas, list[k][2] );
Add( defs, list[k][1] );
Add( weight, i );
od;
lastlength := Length( indices );
od;
npb := RelativeBasisNC( Basis( L ), newbas );
npb!.definitions := defs; npb!.weights := weight;
npb!.isNilpotentBasis := true;
return npb;
end );
######################################################################
##
#P IsNilpotentBasis( <B> )
##
InstallMethod(
IsNilpotentBasis,
"for bases of Lie algebras",
[ IsBasis ],
function( B )
if IsBound( B!.isNilpotentBasis ) then
return B!.isNilpotentBasis;
else
return false;
fi;
end );
######################################################################
##
#A LieNBDefinitions( B )
##
## gets the definitions for a nilpotent basis <B>
##
InstallMethod(
LieNBDefinitions,
"for NB bases of Lie algebras",
[ IsNilpotentBasis ],
function( B )
return B!.definitions;
end );
######################################################################
##
#A LieNBWeights( <B> )
## Gets the weights from a NilpotentBasis
InstallMethod(
LieNBWeights,
"for NB bases of Lie algebras",
[ IsNilpotentBasis ],
function( B )
return B!.weights;
end );
[ Dauer der Verarbeitung: 0.26 Sekunden
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