View source for Cartan-Weyl basis

{{TEX|done}}
''of a finite-dimensional semi-simple complex Lie algebra $\mathfrak g$''

A basis of $\mathfrak g$ consisting of elements of a [[Cartan subalgebra|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|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|Chevalley group]]). The definition of a Cartan–Weyl basis (introduced by H. Weyl in [[#References|[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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Weyl,   "Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch lineare Transformationen I"  ''Math. Z.'' , '''23'''  (1925)  pp. 271–309</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Jacobson,   "Lie algebras" , Interscience  (1962)  ((also: Dover, reprint, 1979))</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N. Bourbaki,   "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley  (1975)  (Translated from French)</TD></TR></table>



====Comments====
See also [[Lie algebra, semi-simple|Lie algebra, semi-simple]] for a description of the special case of a Chevalley basis.

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