#
# file:         semisimple.gap
# purpose:      GAP functions constructing semisimple modular Lie algebras
# created:      pasha jan 4 2005
# modified:     pasha feb 15-17 2005
# modification: continued to work on ToyModularSemisimple1()
# 

# a "toy" model of modular semisimple Lie algebra:
# given a (Lie) algebra L and reduced polynomial ring K[x]/(x^n), 
# constructs L\otimes K[x]/(x^p) + <d/dx>
ToyModularSemisimple := function (L)
	local K, Soc, scSoc, T, p, i, j;
	K := LeftActingDomain(L);
	p := Characteristic(K);
	if p = 0 then
		Error ("the ground field supposed to be of positive characteristic");
	fi;

	Soc := TensorProductOfTwoAlgebras (L, O1);

	# we utilize structure constants in basis constructed in TensorProductOfTwoAlgebras()
	scSoc := StructureConstantsTable (Basis(Soc));

	# now build a semidirect product of Soc and <d/dx> acting on K[x]/(x^n)
	T := EmptySCTable (Dimension(Soc) + 1, Zero(K), "antisymmetric");
	#T := EmptySCTable (Dimension(Soc) + 2, Zero(K), "antisymmetric");

	# basis of L\otimes O_1 + <d/dx> is obtained from basis of L\otimes K[x]/(x^p)...
	for i in [1..Dimension(Soc)] do
		for j in [1..Dimension(Soc)] do
			# this corresponds to the multiplication table in L\otimes K[x]/(x^p)
			T[i][j] := scSoc[i][j];
		od;
	od;

	# ...plus adjoining the last element d/dx
	for i in [1..Dimension(L)] do
		for j in [1..p-1] do
			# this corresponds to [e_i\otimes x^j, d/dx] = j e_i \otimes x^{j-1}
			SetEntrySCTable (T, i+j*Dimension(L), Dimension(Soc) + 1, 
							[j, i+(j-1)*Dimension(L)]);
			##[e_i\otimes x^j, xd/dx] = je_i\otimes x^j
			#SetEntrySCTable (T, i+j*Dimension(L), Dimension(Soc) + 2, 
			#				[j, i+j*Dimension(L)]);
		od;
	od;
		
	## multiplication between d/dx and xd/dx: [d/dx, xd/dx] = -d/dx
	#SetEntrySCTable (T, Dimension(Soc) + 1, Dimension(Soc) + 2, [-1, Dimension(Soc) + 1]);

	return (LieAlgebraByStructureConstants (K, T));
end;




# given a Lie algebra L, and list of derivations of O_1, listders, 
# construct a Lie algebra L\otimes O_1 + <listders>
# (listders supposed to be a basis of algebra of derivations of O_1)
ToyModularSemisimple1 := function (L, listders)
	local K, Soc, scSoc, p, der;
	K := LeftActingDomain(L);
	p := Characteristic(K);
	if p = 0 then
	   Error ("the ground field supposed to be of positive characteristic");
	fi;
	
	Soc := TensorProductOfTwoAlgebras (L, O1);

	# der should be linear span of listder
	der := DerivationLieAlgebra (listders);
	
	# f should be action of der on Soc: der->End(K, Soc) 
	# defined as x\otimes a -> x\otimes d(a)
	#f := LeftModuleHomomorphismByImagesNC (der, End (L, Soc), Basis(der), ?);


	#return (SemidirectProductOfAlgebras (Soc, der, f));
end;


# end of semisimple.gap