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

$[H_{i},H_{j}]=0$ $[H_{i},E_{\alpha }]=\alpha _{i}E_{\alpha }$ Defining the dual root or coroot of $\alpha$ as

$\alpha ^{\vee }={\frac {2\alpha }{(\alpha ,\alpha )}}$ One may perform a change of basis to define

$H_{\alpha _{i}}=(\alpha _{i}^{\vee },H)$ The Cartan integers are

$A_{ij}=(\alpha _{i},\alpha _{j}^{\vee })$ The resulting relations among the generators are the following:

$[H_{\alpha _{i}},H_{\alpha _{j}}]=0$ $[H_{\alpha _{i}},E_{\alpha _{j}}]=A_{ji}E_{\alpha _{j}}$ $[E_{-\alpha _{i}},E_{\alpha _{i}}]=H_{\alpha _{i}}$ $[E_{\beta },E_{\gamma }]=\pm (p+1)E_{\beta +\gamma }$ where in the last relation $p$ is the greatest positive integer such that $\gamma -p\beta$ is a root and we consider $E_{\beta +\gamma }=0$ if $\beta +\gamma$ is not a root.

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