Cartan-Weyl basis

of a finite-dimensional semi-simple complex Lie algebra $\mathfrak g$

A basis of $\mathfrak g$ consisting of elements of a Cartan subalgebra $\mathfrak t$ of $\mathfrak g$ and root vectors $X_\alpha$, $\alpha\in\Delta$, where $\Delta$ is the system of all non-zero roots of $\mathfrak g$ with respect to $\mathfrak t$. The choice of a Cartan–Weyl basis is not unique. A root $\alpha(h)$, $h\in\mathfrak t$, is identified, as a linear form on $\mathfrak t$, with the vector $h_\alpha'\in\mathfrak t$ such that $\alpha(h)=(h_\alpha',h)$, where $(x,y)$ is the Killing form in $\mathfrak g$. Here

$$[h,X_\alpha]=(h_\alpha',h)X_\alpha$$

for each $h\in\mathfrak t$. If $\alpha\in\Delta$, then $-\alpha\in\Delta$ and the root vectors $X_\alpha$ can be chosen such that $[X_\alpha,X_{-\alpha}]=h_\alpha'$. If $\alpha+\beta\in\Delta$, then

$$[X_\alpha,X_\beta]=N_{\alpha\beta}X_{\alpha+\beta},$$

where $N_{\alpha\beta}\neq0$. If $\alpha,\beta,\gamma\in\Delta$, $\alpha+\beta+\gamma=0$, then $N_{\alpha\beta}=N_{\beta\gamma}=N_{\gamma\alpha}$. There exists a normalization of the vectors $X_\alpha$ for which $N_{\alpha\beta}=-N_{-\alpha-\beta}$, where the numbers $N_{\alpha\beta}$ obtained are rational. There exists a normalization of the vectors $X_\alpha$ under which all $N_{\alpha\beta}$ are integers (see Chevalley group). The definition of a Cartan–Weyl basis (introduced by H. Weyl in [1]), as well as everything mentioned above concerning the vectors $X_\alpha$, $h_\alpha'$ and the numbers $N_{\alpha\beta}$, carry over verbatim to the case of an arbitrary finite-dimensional split semi-simple Lie algebra over a field of characteristic zero and its root decomposition with respect to a split Cartan subalgebra.

References

[1]	H. Weyl, "Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch lineare Transformationen I" Math. Z. , 23 (1925) pp. 271–309
[2]	N. Jacobson, "Lie algebras" , Interscience (1962) ((also: Dover, reprint, 1979))
[3]	N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from French)

Comments

See also Lie algebra, semi-simple for a description of the special case of a Chevalley basis.

References

[a1]	J.-P. Serre, "Algèbres de Lie semi-simples complexes" , Benjamin (1966)
[a2]	J.E. Humphreys, "Introduction to Lie algebras and representation theory" , Springer (1972) pp. §5.4
[a3]	R.W. Carter, "Simple groups of Lie type" , Wiley (Interscience) (1972)