########################################################
##  u4_vibron.g     by L.Fortunato and W.A.de Graaf      (2011)
########################################################
##  This is a GAP4 script - 
##  needs SLA package and the files rootsys.g by W.A.de Graaf
##  
##  It comes as auxiliary material to the paper  "Angular momentum non conserving 
##  symmetries in bosonic models"  L.Fortunato, W.A. de Graaf, J.Phys.A : Math. Theor. 44, 145206 (2011)
##
##  u(4) Lie algebra is implemented following the definitions in ch.5 of the book
##  "Symmetry methods in molecules and nuclei" by A.Frank and P.van Isacker,
##  SyG Editores, Mexico (2005)
##
##  Note: set the right path in the first line of code.
########################################################
Read("../Myprogs/rootsys.g");

T:=EmptySCTable(16,0,"antisymmetric");SetEntrySCTable(T,1,10,[1/Sqrt(3), 10]);
SetEntrySCTable(T,1,11,[1/Sqrt(3), 11]);
SetEntrySCTable(T,1,12,[1/Sqrt(3), 12]);
SetEntrySCTable(T,1,13,[-(1/Sqrt(3)), 13]);
SetEntrySCTable(T,1,14,[-(1/Sqrt(3)), 14]);
SetEntrySCTable(T,1,15,[-(1/Sqrt(3)), 15]);
SetEntrySCTable(T,2,3,[1/Sqrt(2), 2]);
SetEntrySCTable(T,2,4,[1/Sqrt(2), 3]);
SetEntrySCTable(T,2,6,[1, 5]);
SetEntrySCTable(T,2,7,[Sqrt(3/2), 6]);
SetEntrySCTable(T,2,8,[Sqrt(3/2), 7]);
SetEntrySCTable(T,2,9,[1, 8]);
SetEntrySCTable(T,2,11,[1/Sqrt(2), 10]);
SetEntrySCTable(T,2,12,[1/Sqrt(2), 11]);
SetEntrySCTable(T,2,14,[1/Sqrt(2), 13]);
SetEntrySCTable(T,2,15,[1/Sqrt(2), 14]);
SetEntrySCTable(T,3,4,[1/Sqrt(2), 4]);
SetEntrySCTable(T,3,5,[-Sqrt(2), 5]);
SetEntrySCTable(T,3,6,[-(1/Sqrt(2)), 6]);
SetEntrySCTable(T,3,8,[1/Sqrt(2), 8]);
SetEntrySCTable(T,3,9,[Sqrt(2), 9]);
SetEntrySCTable(T,3,10,[-(1/Sqrt(2)), 10]);
SetEntrySCTable(T,3,12,[1/Sqrt(2), 12]);
SetEntrySCTable(T,3,13,[-(1/Sqrt(2)), 13]);
SetEntrySCTable(T,3,15,[1/Sqrt(2), 15]);
SetEntrySCTable(T,4,5,[-1, 6]);
SetEntrySCTable(T,4,6,[-Sqrt(3/2), 7]);
SetEntrySCTable(T,4,7,[-Sqrt(3/2), 8]);
SetEntrySCTable(T,4,8,[-1, 9]);
SetEntrySCTable(T,4,10,[-(1/Sqrt(2)), 11]);
SetEntrySCTable(T,4,11,[-(1/Sqrt(2)), 12]);
SetEntrySCTable(T,4,13,[-(1/Sqrt(2)), 14]);
SetEntrySCTable(T,4,14,[-(1/Sqrt(2)), 15]);
SetEntrySCTable(T,5,8,[-1, 2]);
SetEntrySCTable(T,5,9,[-Sqrt(2), 3]);
SetEntrySCTable(T,5,12,[1, 10]);
SetEntrySCTable(T,5,15,[-1, 13]);
SetEntrySCTable(T,6,7,[Sqrt(3/2), 2]);
SetEntrySCTable(T,6,8,[1/Sqrt(2), 3]);
SetEntrySCTable(T,6,9,[-1, 4]);
SetEntrySCTable(T,6,11,[-(1/Sqrt(2)), 10]);
SetEntrySCTable(T,6,12,[1/Sqrt(2), 11]);
SetEntrySCTable(T,6,14,[1/Sqrt(2), 13]);
SetEntrySCTable(T,6,15,[-(1/Sqrt(2)), 14]);
SetEntrySCTable(T,7,8,[Sqrt(3/2), 4]);
SetEntrySCTable(T,7,10,[1/Sqrt(6), 10]);
SetEntrySCTable(T,7,11,[-Sqrt(2/3), 11]);
SetEntrySCTable(T,7,12,[1/Sqrt(6), 12]);
SetEntrySCTable(T,7,13,[-(1/Sqrt(6)), 13]);
SetEntrySCTable(T,7,14,[Sqrt(2/3), 14]);
SetEntrySCTable(T,7,15,[-(1/Sqrt(6)), 15]);
SetEntrySCTable(T,8,10,[1/Sqrt(2), 11]);
SetEntrySCTable(T,8,11,[-(1/Sqrt(2)), 12]);
SetEntrySCTable(T,8,13,[-(1/Sqrt(2)), 14]);
SetEntrySCTable(T,8,14,[1/Sqrt(2), 15]);
SetEntrySCTable(T,9,10,[1, 12]);
SetEntrySCTable(T,9,13,[-1, 15]);
SetEntrySCTable(T,10,13,[1, 5]);
SetEntrySCTable(T,10,14,[-(1/Sqrt(2)), 2, 1/Sqrt(2), 6]);
SetEntrySCTable(T,10,15,[1/Sqrt(3), 1, -(1/Sqrt(2)), 3, 1/Sqrt(6), 7, -1, 16]);
SetEntrySCTable(T,10,16,[1, 10]);
SetEntrySCTable(T,11,13,[1/Sqrt(2), 2, 1/Sqrt(2), 6]);
SetEntrySCTable(T,11,14,[-(1/Sqrt(3)), 1, Sqrt(2/3), 7, 1, 16]);
SetEntrySCTable(T,11,15,[-(1/Sqrt(2)), 4, 1/Sqrt(2), 8]);
SetEntrySCTable(T,11,16,[1, 11]);
SetEntrySCTable(T,12,13,[1/Sqrt(3), 1, 1/Sqrt(2), 3, 1/Sqrt(6), 7, -1, 16]);
SetEntrySCTable(T,12,14,[1/Sqrt(2), 4, 1/Sqrt(2), 8]);
SetEntrySCTable(T,12,15,[1, 9]);
SetEntrySCTable(T,12,16,[1, 12]);
SetEntrySCTable(T,13,16,[-1, 13]);
SetEntrySCTable(T,14,16,[-1, 14]);
SetEntrySCTable(T,15,16,[-1, 15]);

Print("\n");
F:=DefaultField(Sqrt(2),Sqrt(3));
Print("F=",F,"\n");
L:=AlgebraByStructureConstants(F,T);
Print("L=",L,"\n\n");
Print("Jacobi-> ",TestJacobi(T),"\n" );
B:=Basis(L);
Print("B=",B,"\n\n");

g:=KillingMatrix(Basis(L));

Print(g,"\n");
Print("det=",Determinant(g),"\n");

LM:=LeviMalcevDecomposition(L);
Print("Levi-Malcev Decomposition:",LM,"\n");
bs:=BasisVectors(Basis(LM[1]));
br:=BasisVectors(Basis(LM[2]));
Print("LM[2] -Basis of radical subalg of dim ",Dimension(LM[2]),": ",br, "\n");
Ro:=rootsystem(LM[1]);  
Print("Ro=",Ro,"\n");
sst:=CartanType(CartanMatrix(Ro));
Print("LM[1] -Basis of semisim subalg of dim ",Dimension(LM[1]),": ",bs, " of type ",sst,"\n" );

C:=CartanSubalgebra(LM[1]);