########################################################
##  u3.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(3) 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(9,0,"antisymmetric");

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,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,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,5,8,[-1, 2]);
SetEntrySCTable(T,5,9,[-Sqrt(2), 3]);
SetEntrySCTable(T,6,7,[Sqrt(3/2), 2]);
SetEntrySCTable(T,6,8,[1/Sqrt(2), 3]);
SetEntrySCTable(T,6,9,[-1, 4]);
SetEntrySCTable(T,7,8,[Sqrt(3/2), 4]);

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" );
Ba:=Basis(L);
Print("Ba=",Ba,"\n\n");

g:=KillingMatrix(Basis(L));
Print(g,"\n");
Print("det=",Determinant(g),"\n\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");
sst:=CartanType(CartanMatrix(rootsystem(LM[1])));
Print("LM[1] -Basis of semisim subalg of dim ",Dimension(LM[1]),": ",bs, " of type ",sst,"\n" );