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