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
. The Cartan-Weyl basis may be written as
![{\displaystyle [H_{i},H_{j}]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b8d32f1a80df904b21d33f702337372bc13c8c3)
![{\displaystyle [H_{i},E_{\alpha }]=\alpha _{i}E_{\alpha ))](https://wikimedia.org/api/rest_v1/media/math/render/svg/659a4275515d0b5ac4fa95832a5bdd370bc5f767)
Defining the dual root or coroot of
as
![{\displaystyle \alpha ^{\vee }={\frac {2\alpha }{(\alpha ,\alpha )))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a44bbff7b570e02298e8909bac53be32650a463e)
One may perform a change of basis to define
![{\displaystyle H_{\alpha _{i))=(\alpha _{i}^{\vee },H)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cdc32eaa8908268b6cf01f4b4b6600ea75eba8b8)
The Cartan integers are
![{\displaystyle A_{ij}=(\alpha _{i},\alpha _{j}^{\vee })}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e709535ffefac0373ae6a09183303b8af7e0f645)
The resulting relations among the generators are the following:
![{\displaystyle [H_{\alpha _{i)),H_{\alpha _{j))]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/212ee9ac11479ce006603aa5fc01efac3778b892)
![{\displaystyle [H_{\alpha _{i)),E_{\alpha _{j))]=A_{ji}E_{\alpha _{j))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/301f73defa250441c4e127138a045e1f40d4bc07)
![{\displaystyle [E_{-\alpha _{i)),E_{\alpha _{i))]=H_{\alpha _{i))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/467debba4ee2df5b67afe9136238c8ba2654bc10)
![{\displaystyle [E_{\beta },E_{\gamma }]=\pm (p+1)E_{\beta +\gamma ))](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc3a1a2a23dd693bc2fadd12ffa8d0d31117ad7d)
where in the last relation
is the greatest positive integer such that
is a root and we consider
if
is not a root.
For determining the sign in the last relation one fixes an ordering of roots which respects addition, i.e., if
then
provided that all four are roots. We then call
an extraspecial pair of roots if they are both positive and
is minimal among all
that occur in pairs of positive roots
satisfying
. The sign in the last relation can be chosen arbitrarily whenever
is an extraspecial pair of roots. This then determines the signs for all remaining pairs of roots.