[GAP Forum] semisimple type and chevalley basis

[degraaf at science.unitn.it]

Dear Tsai,

The problem is that your Lie algebra does not
have a split Cartan subalgebra (ie the eigenvalues
of the adjoint map are not in the base field).
For this reason GAP cannot construct the root
system, and, since ChevalleyBasis uses this,
it also runs into an error.
The function SemiSimpleType recognizes the Lie
algebra without computing the root system.
I do agree that the error message could be more
informative.

Best wishes,

Willem de Graaf



Dear gap-forum,

I define a lie algebra which I know to be equivalent to type "A1".
I was hoping to have gap calculate a chavalley basis for this, but
this doesn't seem to be possible; even though gap recognizes the
algebra as "A1" (using SemiSimpleType). Is there a way around this?
Thanks,
R.N.

mats:=
[ [ [ 0, -1, 0 ], [ 1, 0, 0 ], [ 0, 0, 0 ] ],
  [ [ 0, 0, -1 ], [ 0, 0, 0 ], [ 1, 0, 0 ] ],
  [ [ 0, 0, 0 ], [ 0, 0, -1 ], [ 0, 1, 0 ] ] ];

alg:=LieAlgebra(Rationals,mats);

gap recognizes the algebra as "A1"

gap> SemiSimpleType(alg);
"A1"

but it has trouble with its chevalley basis

gap> ChevalleyBasis(alg);
Error, no method found! For debugging hints type ?Recovery from NoMethodFound
Error, no 1st choice method found for `PositiveRoots' on 1 arguments  
called fr\
om
PositiveRoots( R ) called from
<function>( <arguments> ) called from read-eval-loop
Entering break read-eval-print loop ...
you can 'quit;' to quit to outer loop, or
you can 'return;' to continue
brk> quit;





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