[GAP Forum] Root system of semi-simple Lie algebra G2
Alexander Konovalov
alexk at mcs.st-andrews.ac.uk
Mon Dec 9 21:43:33 GMT 2013
Dear Georg and all,
Just to add that CoReLG and three other packages extending GAP's
functionality for Lie algebras, namely LieRing, LiePRing and SLA,
are included in GAP 4.7.2 distribution which was announced on Friday,
and I'd like to encourage you to get CoReLG via upgrading to
GAP 4.7.2 instead of trying to install it under GAP 4.6.4 which is
dated May 2013.
Best wishes
Alexander
On 9 Dec 2013, at 21:03, Willem de Graaf <degraaf at science.unitn.it> wrote:
> 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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