[GAP Forum] Exceptional compact Lie algebras

[mim_ at op.pl]

Hello,

I am trying to obtain compact exceptional Lie algebras: e7,e8 in matrix represantation. I have this already created for f4 and e6 as matrices 27*27 real and complex.

For e7 it is 56-dimensional, and it is quaternionic, so I would like to have inclusion e7 in sp(28). 28 contains 27 dimensions from h3O and additional dimension for DerH. I have tried Adams way but no luck. I am going to try Freudenthal "Beziehungen..." now. 

For e8 I have obtained compact version from default e8 GAP algebra by manipulating the basis and multiplying by i=Sqrt(-1). The adjoint representation gives 248*248 matrices. However I have difficulties to find spin(16) there. I can try random subalgebras generated by the 6-10 base elements and see the results. My idea is to look at them as 31*8 matrices, 31=3*8+3+4, so we have 27 dimensions from h3O and additional 4 dimensions for DerO.

1. Do you have any ideas for building matrix representations of compact e7, e8 in dimensions 56,248 respectively ? 

2. The root system in GAP is different to the one presented on wikipedia for F4, E6,E7,E8. Is there way to take wikipedia roots and build lie algebra from it ?

3. When I tried to do Derivations from 54 dimensional algebra C*h3O there is out-of-memory error which close the GAP session. Any workaround ? 

Regards,
Marek Mitros