In mathematics, a Chevalley basis for a simple complex Lie algebra is a basis constructed by Claude Chevalley with the property that all structure constants are integers. Chevalley used these bases to construct analogues of Lie groups over finite fields, called Chevalley groups. The Chevalley basis is the Cartan-Weyl basis, but with a different normalization.
The generators of a Lie group are split into the generators H and E indexed by simple roots and their negatives
\pm\alphai
[Hi,Hj]=0
[Hi,E\alpha]=\alphaiE\alpha
Defining the dual root or coroot of
\alpha
\alpha\vee=
2\alpha | |
(\alpha,\alpha) |
where
( ⋅ , ⋅ )
H | |
\alphai |
\vee, | |
=(\alpha | |
i |
H)
The Cartan integers are
Aij=(\alphai,\alpha
\vee) | |
j |
[H | |
\alphai |
,H | |
\alphaj |
]=0
[H | |
\alphai |
,E | |
\alphaj |
]=Aji
E | |
\alphaj |
[E | |
-\alphai |
,E | |
\alphai |
]=
H | |
\alphai |
[E\beta,E\gamma]=\pm(p+1)E\beta+\gamma
p
\gamma-p\beta
E\beta=0
\beta+\gamma
For determining the sign in the last relation one fixes an ordering of roots which respects addition, i.e., if
\beta\prec\gamma
\beta+\alpha\prec\gamma+\alpha
(\beta,\gamma)
\beta
\beta0
(\beta0,\gamma0)
\beta0+\gamma0=\beta+\gamma
(\beta,\gamma)