ChevalleyBasis Details - Maple Help
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Chevalley Basis Details

 

The details for the construction of the Chevalley basis are as follows. Let be a real, split semi-simple Lie algebra. Start with a basis for , where , is a basis for a Cartan subalgebra, and where , (the positive roots), gives a root space decomposition for . By definition of a split, semi-simple Lie algebra, the root vectors are all real. Let  be the Killing form. Scale the vectorssuch that and set . Scale the vectors again (preserving ) so that the structure equations

hold. Let be the simple roots, and set

.

This fixes the vectors  in the Chevalley basis . Write

 .

We need to make one final scaling of the vectors ,for. We calculate the structure constants , for and and generate the system of quadratic equations

 .

Here is the largest positive integer such that is not a root. Put for and solve for the remaining . Finally set  and put

  and  for .

This completes the construction of the Chevalley basis ' . We have

for all , where the matrix  is the Cartan matrix for  and, also,

 where .

Note that in the Chevalley basis all the structure constants are integers and that the transformation ,  is a Lie algebra automorphism.

 

See N. Bourbaki, Lie Groups and Lie Algebras, Chapters 7-9, Section 4 for additional details.


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