#
# file:         der.gap
# purpose:      GAP functions related to derivations
# created:      pasha jan 13 2005
# modified:     pasha feb 17,19 2005
# modification: finished LieDerivationAlgebra()
#


# tests whether given mapping D:R->S is a derivation
# (R and S supposed to be algebras); returns true of false
TestDerivation := function (D)
	local R, B, i, j;
	R := Range(D);
	B := Basis(R);
	for i in [1..Dimension(R)] do
		for j in [1..Dimension(R)] do
			if Image (D, B[i]*B[j]) <> Image (D, B[i])*B[j] + B[i]*Image (D, B[j]) then
				return (false);
			fi;
		od;
	od;
	return (true);
end;


# get Lie algebra generated as a vector space
# by the given list of derivations of (the same) algebra;
# returns false if the given list is not a basis of a Lie algebra of derivations
# (i.e. either it is not linearly independent or not closed under Lie multiplication)
LieDerivationAlgebra := function (listder)
	local K, V, T, i, j, k, cf, list;
	
	# TODO: check that range and source of all derivations is the same
	K := LeftActingDomain (Range(listder[1]));
	V := VectorSpace (K, listder, "basis");
	T := EmptySCTable (Length(listder), Zero(K));

	for i in [1..Length(listder)] do
		for j in [1..i-1] do
			cf := Coefficients (Basis(V), listder[i]*listder[j] - listder[j]*listder[i]);
			list := [];
			for k in [1..Length(cf)] do
				if cf[k] <> 0 then
					Add (list, cf[k]);
					Add (list, k);
				fi;
			od;
			SetEntrySCTable (T, i, j, list);
		od;
	od;
	
	return (LieAlgebraByStructureConstants (K, T));
end;


# end of der.gap