Chevalley basis

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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 generators of a Lie group are split into the generators H and E indexed by simple roots and their negatives \pm\alpha_i. The relations among the generators are the following:

[H_{\alpha_i},H_{\alpha_j}]=0
[H_{\alpha_i},E_{\alpha_j}]=\alpha_j(H_{\alpha_i}) E_{\alpha_j}
[E_{-\alpha_i},E_{\alpha_i}] = H_{\alpha_i}
[E_{\beta},E_{\gamma}]=\pm(p+1)E_{\beta+\gamma}

where the last relation is imposed only if \beta + \gamma is a root and p is the greatest positive integer such that γ − pβ is a root. The sign in the relation can be fixed arbitrarily.

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