Chevalley Basis

A Chevalley basis of a complex semisimple Lie algebra \(\mathfrak g\) is a basis \[ \{H_\alpha : \alpha \in \Delta\} \cup \{E_\alpha : \alpha \in \Phi\} \] (indexed by simple roots \(\Delta\) and all roots \(\Phi\)) satisfying the integer structure constant conditions:

\[ [H_\alpha, H_\beta] = 0, \quad [H_\alpha, E_\beta] = \langle \beta, \alpha^\vee \rangle\, E_\beta, \] \[ [E_\alpha, E_{-\alpha}] = H_\alpha, \quad [E_\alpha, E_\beta] = N_{\alpha,\beta}\, E_{\alpha+\beta} \quad (\alpha + \beta \in \Phi), \] where \(N_{\alpha,\beta} \in \mathbb{Z} \setminus \{0\}\) and \(N_{\alpha,\beta} = -N_{-\alpha,-\beta}\).

Key properties

Integer structure constants: all \(N_{\alpha,\beta} \in \mathbb{Z}\), so the \(\mathbb{Z}\)-span \[ \mathfrak g_{\mathbb{Z}} = \bigoplus_{\alpha \in \Delta} \mathbb{Z} H_\alpha \oplus \bigoplus_{\alpha \in \Phi} \mathbb{Z} E_\alpha \] is a Lie algebra over \(\mathbb{Z}\) (a Chevalley \(\mathbb{Z}\)-form).
Base change: \(\mathfrak g_{\mathbb{Z}} \otimes_{\mathbb{Z}} k\) is a Lie algebra over any field \(k\), and the structure constants remain the same integers modulo \(\mathrm{char}(k)\).
Sign ambiguity: The basis is not unique. Different Chevalley bases for the same root system differ by signs \(E_\alpha \mapsto \pm E_\alpha\). However, the isomorphism type of the resulting integral form \(\mathfrak g_{\mathbb{Z}}\) is unique.
Existence: Chevalley proved existence for all complex semisimple Lie algebras by explicit computation using Serre's presentation.

Structure constants \(N_{\alpha,\beta}\)

For non-proportional roots \(\alpha, \beta \in \Phi\) with \(\alpha + \beta \in \Phi\), define the \(\alpha\)-string through \(\beta\) as \[ \beta - q\alpha, \ldots, \beta, \ldots, \beta + p\alpha \quad (q,p \geq 0, \; q + p = -\langle \beta, \alpha^\vee \rangle + 2p). \] Then \[ N_{\alpha,\beta} = \pm(q+1), \] where the sign is fixed by a global sign convention on the basis.

Role in Chevalley group construction

The Chevalley basis is the essential input for the Chevalley group construction:

The elements \(H_\alpha, E_\alpha\) define a faithful representation on a free \(\mathbb{Z}\)-module \(L\) (a lattice in the representation).
The root subgroup elements are formal exponentials \[ x_\alpha(t) = \exp(t E_\alpha) = \sum_{k=0}^\infty \frac{t^k}{k!} E_\alpha^k. \] Since \(E_\alpha\) is nilpotent on \(L\), the sum is finite and \(x_\alpha(t)\) acts on \(L \otimes R\) for any ring \(R\).
The map \(t \mapsto x_\alpha(t)\) is a group scheme morphism \(\mathbb{G}_a \to \mathrm{GL}(L)\), defining the root subgroup \(U_\alpha\).
The commutator \([x_\alpha(s), x_\beta(t)]\) is computed by the Chevalley commutator formula, whose coefficients involve the \(N_{\alpha,\beta}\) and are integers.

Examples

Root system	\(\mathfrak g\)	Basis elements
\(A_1\)	\(\mathfrak{sl}_2\)	\(H, E, F\) with \([H,E]=2E, [H,F]=-2F, [E,F]=H\)
\(A_{n-1}\)	\(\mathfrak{sl}_n\)	\(E_{ij}\) (matrix units) for \(i \ne j\), \(H_i = E_{ii} - E_{i+1,i+1}\)
\(G_2\)	Exceptional	14-dim basis with specific integer structure constants

References