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" ); _______________________________________________ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum
