Chevalley basis - Knowlpedia

Let $\mathfrak g$ be a split semisimple Lie algebra over a characteristic-zero field, with split Cartan subalgebra $\mathfrak t$ , root system $\Phi$ , and simple roots $\Delta=\{\alpha_1,\ldots,\alpha_\ell\}$ . A Chevalley basis is

\{h_1,\ldots,h_\ell\}\cup\{e_\alpha\}_{\alpha\in\Phi},

where $h_i$ represents the simple coroot and $e_\alpha$ spans $\mathfrak g_\alpha$ , normalized so that

[h_i,e_\alpha] = \langle\alpha,\alpha_i^\vee\rangle e_\alpha, \qquad [e_\alpha,e_{-\alpha}]=h_\alpha,

and

[e_\alpha,e_\beta] = N_{\alpha,\beta}e_{\alpha+\beta}

when $\alpha+\beta$ is a root, with $N_{\alpha,\beta}\in\mathbb Z$ .

Root-string normalization

The root-vector signs can be chosen so that

N_{\alpha,\beta}=\pm(p+1),

where $p$ is the largest nonnegative integer such that $\beta-p\alpha\in\Phi$ . The sign choices are not canonical, but the existence of integral structure constants is.

Integral Lie form

The $\mathbb Z$ -span

\mathfrak g_{\mathbb Z} = \bigoplus_i\mathbb Zh_i \oplus \bigoplus_{\alpha\in\Phi}\mathbb Ze_\alpha

is closed under the bracket. It is the Chevalley Lie form. Constructing integral representations and a reductive group scheme additionally uses divided powers and the full root datum; see the integral-model page.

Role in the letter

Root-vector normalization rigidifies the pinning and gives integral structures from which almost-all local integral models and spherical Hecke algebras can be obtained.

References

Claude Chevalley, “Sur certains groupes simples,” Tohoku Mathematical Journal 7 (1955), 14–66.
Robert Steinberg, Lectures on Chevalley Groups, AMS, 2016.