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In mathematics, aChevalley basis for asimplecomplexLie algebra is abasis constructed byClaude Chevalley with the property that allstructure constants are integers. Chevalley used these bases to construct analogues ofLie groups overfinite fields, calledChevalley groups. The Chevalley basis is theCartan-Weyl basis, but with a different normalization.

The generators of a Lie group are split into the generatorsH andE indexed by simpleroots 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 thedual root orcoroot of${\displaystyle \alpha }$ as

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

where${\displaystyle (\cdot ,\cdot )}$ is the euclidean inner product. One may perform a change of basis to define

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

TheCartan 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${\displaystyle (\beta ,\gamma )}$ anextraspecial 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.

• 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.
• Tits, Jacques (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.doi:10.1007/BF02684801.MR 0214638.Zbl 0145.25804.

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