# Chevalley basis

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 ${\displaystyle \pm \alpha _{i}}$. The Cartan-Weyl basis may be written as

${\displaystyle [H_{i},H_{j}]=0}$
${\displaystyle [H_{i},E_{\alpha }]=\alpha _{i}E_{\alpha }}$

Defining the dual root or coroot of ${\displaystyle \alpha }$ as

${\displaystyle \alpha ^{\vee }={\frac {2\alpha }{(\alpha ,\alpha )}}}$

One may perform a change of basis to define

${\displaystyle H_{\alpha _{i}}=(\alpha _{i}^{\vee },H)}$

The Cartan integers are

${\displaystyle A_{ij}=(\alpha _{i},\alpha _{j}^{\vee })}$

The resulting relations among the generators are the following:

${\displaystyle [H_{\alpha _{i}},H_{\alpha _{j}}]=0}$
${\displaystyle [H_{\alpha _{i}},E_{\alpha _{j}}]=A_{ji}E_{\alpha _{j}}}$
${\displaystyle [E_{-\alpha _{i}},E_{\alpha _{i}}]=H_{\alpha _{i}}}$
${\displaystyle [E_{\beta },E_{\gamma }]=\pm (p+1)E_{\beta +\gamma }}$

where in the last relation ${\displaystyle p}$ is the greatest positive integer such that ${\displaystyle \gamma -p\beta }$ is a root and we consider ${\displaystyle E_{\beta +\gamma }=0}$ if ${\displaystyle \beta +\gamma }$ is not a root.

For determining the sign in the last relation one fixes an ordering of roots which respects addition, i.e., if ${\displaystyle \beta \prec \gamma }$ then ${\displaystyle \beta +\alpha \prec \gamma +\alpha }$ provided that all four are roots. We then call ${\displaystyle (\beta ,\gamma )}$ an extraspecial pair of roots if they are both positive and ${\displaystyle \beta }$ is minimal among all ${\displaystyle \beta _{0}}$ that occur in pairs of positive roots ${\displaystyle (\beta _{0},\gamma _{0})}$ satisfying ${\displaystyle \beta _{0}+\gamma _{0}=\beta +\gamma }$. The sign in the last relation can be chosen arbitrarily whenever ${\displaystyle (\beta ,\gamma )}$ is an extraspecial pair of roots. This then determines the signs for all remaining pairs of roots.

## References

• Carter, Roger W. (1993). Finite Groups of Lie Type: Conjugacy Classes and Complex Characters. Wiley Classics Library. Chichester: Wiley. ISBN 978-0-471-94109-5.
• Chevalley, Claude (1955). "Sur certains groupes simples". Tohoku Mathematical Journal (in French). 7 (1–2): 14–66. doi:10.2748/tmj/1178245104. MR 0073602. Zbl 0066.01503.
• Jacques, Tits (1966). "Sur les constantes de structure et le théorème d'existence des algèbres de Lie semi-simples". Publications Mathématiques de l'IHÉS (in French). 31: 21–58. MR 0214638. Zbl 0145.25804.