[GAP Forum] Root system of semi-simple Lie algebra G2
Willem de Graaf
degraaf at science.unitn.it
Mon Dec 9 21:03:45 GMT 2013
Dear Georg,
The problem is that the Cartan subalgebra that has been computed is not
split over the rationals. It is split over Q(i).
However, GAP does not take advantage of that. Recently we have written
a package, called corelg, which does manage to compute the root system
in this case (http://www.science.unitn.it/~corelg/):
gap> L := LieAlgebraByStructureConstants( CF(4), T );
<Lie algebra of dimension 14 over GaussianRationals>
gap> RootSystem(L);
<root system of rank 2>
Best wishes,
Willem de Graaf
On Sun, Dec 8, 2013 at 11:43 PM, Georg Beyerle <mail at gbeyerle.de> wrote:
> 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" );
>
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