[GAP Forum] Root system of semi-simple Lie algebra G2

[Georg Beyerle]

Hello,

the following table of structure constants (see below) appears to define
a semi-simple Lie algebra. However, the call to RootSystem() fails:

   GAPInfo.Version = 4.6.4
   GAPInfo.Architecture = i586-suse-linux-gnu-gcc-default32
   Dimension( L ) = 14
   SemiSimpleType( L ) : G2
   Determinant( KillingMatrix( Basis( L ))) = 9618527719784448
   CartanSubalgebra( L ) = Algebra( Rationals, [ v.1, v.8 ] )
   #I  the Cartan subalgebra of <L> in not split
   RootSystem( L ) = fail

What am I missing?

Thanks
Georg


   Print( "GAPInfo.Version = ", GAPInfo.Version, "\n" );
   Print( "GAPInfo.Architecture = ", GAPInfo.Architecture, "\n" );
   SetInfoLevel( InfoAlgebra , 2 );
   T:= EmptySCTable( 14, 0, "antisymmetric" );;
   SetEntrySCTable( T, 1, 2, [  -1 , 3 , -1 , 10 ] );;
   SetEntrySCTable( T, 1, 3, [  1 , 2 , 1 , 9 ] );;
   SetEntrySCTable( T, 1, 4, [  1 , 5 , -1 , 12 ] );;
   SetEntrySCTable( T, 1, 5, [  -1 , 4 , -1 , 11 ] );;
   SetEntrySCTable( T, 1, 6, [  1 , 14 ] );;
   SetEntrySCTable( T, 1, 7, [  1 , 6 , 1 , 13 ] );;
   SetEntrySCTable( T, 1, 9, [  1 , 10 ] );;
   SetEntrySCTable( T, 1, 10, [  -1 , 9 ] );;
   SetEntrySCTable( T, 1, 11, [  1 , 12 ] );;
   SetEntrySCTable( T, 1, 12, [  -1 , 11 ] );;
   SetEntrySCTable( T, 1, 13, [  1 , 14 ] );;
   SetEntrySCTable( T, 1, 14, [  -2 , 6 , -2 , 13 ] );;
   SetEntrySCTable( T, 2, 3, [  -1 , 1 , -1 , 8 ] );;
   SetEntrySCTable( T, 2, 4, [  -1 , 6 , -1 , 13 ] );;
   SetEntrySCTable( T, 2, 5, [  -1 , 7 , -1 , 14 ] );;
   SetEntrySCTable( T, 2, 6, [  -1 , 11 ] );;
   SetEntrySCTable( T, 2, 7, [  1 , 5 , -1 , 12 ] );;
   SetEntrySCTable( T, 2, 8, [  -1 , 10 ] );;
   SetEntrySCTable( T, 2, 10, [  1 , 8 ] );;
   SetEntrySCTable( T, 2, 11, [  1 , 6 ] );;
   SetEntrySCTable( T, 2, 12, [  1 , 7 , 1 , 14 ] );;
   SetEntrySCTable( T, 2, 13, [  1 , 4 , 1 , 11 ] );;
   SetEntrySCTable( T, 2, 14, [  1 , 5 , -1 , 12 ] );;
   SetEntrySCTable( T, 3, 4, [  -1 , 7 ] );;
   SetEntrySCTable( T, 3, 5, [  -2 , 6 ] );;
   SetEntrySCTable( T, 3, 6, [  2 , 5 ] );;
   SetEntrySCTable( T, 3, 7, [  1 , 4 ] );;
   SetEntrySCTable( T, 3, 8, [  1 , 9 ] );;
   SetEntrySCTable( T, 3, 9, [  -1 , 8 ] );;
   SetEntrySCTable( T, 3, 11, [  1 , 7 , 1 , 14 ] );;
   SetEntrySCTable( T, 3, 12, [  -1 , 6 ] );;
   SetEntrySCTable( T, 3, 13, [  -1 , 5 ] );;
   SetEntrySCTable( T, 3, 14, [  -1 , 4 , -1 , 11 ] );;
   SetEntrySCTable( T, 4, 5, [  -1 , 8 ] );;
   SetEntrySCTable( T, 4, 6, [  1 , 9 ] );;
   SetEntrySCTable( T, 4, 7, [  -2 , 3 , -2 , 10 ] );;
   SetEntrySCTable( T, 4, 8, [  1 , 5 ] );;
   SetEntrySCTable( T, 4, 9, [  -1 , 6 ] );;
   SetEntrySCTable( T, 4, 10, [  1 , 7 ] );;
   SetEntrySCTable( T, 4, 12, [  -1 , 1 , -1 , 8 ] );;
   SetEntrySCTable( T, 4, 13, [  -1 , 2 , -1 , 9 ] );;
   SetEntrySCTable( T, 4, 14, [  1 , 3 , 1 , 10 ] );;
   SetEntrySCTable( T, 5, 6, [  -2 , 3 ] );;
   SetEntrySCTable( T, 5, 7, [  1 , 9 ] );;
   SetEntrySCTable( T, 5, 8, [  -1 , 4 ] );;
   SetEntrySCTable( T, 5, 9, [  -1 , 7 ] );;
   SetEntrySCTable( T, 5, 10, [  -1 , 6 ] );;
   SetEntrySCTable( T, 5, 11, [  -1 , 1 , -1 , 8 ] );;
   SetEntrySCTable( T, 5, 13, [  1 , 3 ] );;
   SetEntrySCTable( T, 5, 14, [  -1 , 2 , -1 , 9 ] );;
   SetEntrySCTable( T, 6, 7, [  1 , 8 ] );;
   SetEntrySCTable( T, 6, 8, [  -1 , 7 ] );;
   SetEntrySCTable( T, 6, 9, [  1 , 4 ] );;
   SetEntrySCTable( T, 6, 10, [  1 , 5 ] );;
   SetEntrySCTable( T, 6, 11, [  -1 , 2 ] );;
   SetEntrySCTable( T, 6, 12, [  1 , 3 ] );;
   SetEntrySCTable( T, 6, 14, [  1 , 1 ] );;
   SetEntrySCTable( T, 7, 8, [  1 , 6 ] );;
   SetEntrySCTable( T, 7, 9, [  1 , 5 ] );;
   SetEntrySCTable( T, 7, 10, [  -1 , 4 ] );;
   SetEntrySCTable( T, 7, 11, [  -1 , 3 , -1 , 10 ] );;
   SetEntrySCTable( T, 7, 12, [  -1 , 2 , -1 , 9 ] );;
   SetEntrySCTable( T, 7, 13, [  1 , 1 , 1 , 8 ] );;
   SetEntrySCTable( T, 8, 9, [  -2 , 10 ] );;
   SetEntrySCTable( T, 8, 10, [  2 , 9 ] );;
   SetEntrySCTable( T, 8, 11, [  1 , 12 ] );;
   SetEntrySCTable( T, 8, 12, [  -1 , 11 ] );;
   SetEntrySCTable( T, 8, 13, [  -1 , 7 , -1 , 14 ] );;
   SetEntrySCTable( T, 8, 14, [  1 , 6 , 1 , 13 ] );;
   SetEntrySCTable( T, 9, 10, [  -2 , 8 ] );;
   SetEntrySCTable( T, 9, 11, [  -1 , 6 , -1 , 13 ] );;
   SetEntrySCTable( T, 9, 12, [  -1 , 14 ] );;
   SetEntrySCTable( T, 9, 13, [  1 , 4 , 1 , 11 ] );;
   SetEntrySCTable( T, 9, 14, [  1 , 12 ] );;
   SetEntrySCTable( T, 10, 11, [  -1 , 14 ] );;
   SetEntrySCTable( T, 10, 12, [  1 , 6 , 1 , 13 ] );;
   SetEntrySCTable( T, 10, 13, [  1 , 5 , -1 , 12 ] );;
   SetEntrySCTable( T, 10, 14, [  1 , 11 ] );;
   SetEntrySCTable( T, 11, 12, [  2 , 1 , 2 , 8 ] );;
   SetEntrySCTable( T, 11, 13, [  -1 , 2 , -1 , 9 ] );;
   SetEntrySCTable( T, 11, 14, [  -1 , 10 ] );;
   SetEntrySCTable( T, 12, 13, [  1 , 3 , 1 , 10 ] );;
   SetEntrySCTable( T, 12, 14, [  -1 , 9 ] );;
   SetEntrySCTable( T, 13, 14, [  1 , 1 ] );;

   L := LieAlgebraByStructureConstants( Rationals, T );
   Print( "Dimension( L ) = ", Dimension( L ), "\n" );
   typ := SemiSimpleType( L );
   Print( "SemiSimpleType( L ) : ", typ, "\n" );
   detKM := Determinant( KillingMatrix( Basis( L )));
   Print( "Determinant( KillingMatrix( Basis( L ))) = ", detKM, "\n" );
   H := CartanSubalgebra( L );
   Print( "CartanSubalgebra( L ) = ", H, "\n" );
   R := RootSystem( L );
   Print( "RootSystem( L ) = ", R, "\n" );