View source for Cartan-Weyl basis

''of a finite-dimensional semi-simple complex Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020580/c0205801.png" />''

A basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020580/c0205802.png" /> consisting of elements of a [[Cartan subalgebra|Cartan subalgebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020580/c0205803.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020580/c0205804.png" /> and root vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020580/c0205805.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020580/c0205806.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020580/c0205807.png" /> is the system of all non-zero roots of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020580/c0205808.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020580/c0205809.png" />. The choice of a Cartan–Weyl basis is not unique. A root <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020580/c02058010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020580/c02058011.png" />, is identified, as a linear form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020580/c02058012.png" />, with the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020580/c02058013.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020580/c02058014.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020580/c02058015.png" /> is the [[Killing form|Killing form]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020580/c02058016.png" />. Here

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020580/c02058017.png" /></td> </tr></table>

for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020580/c02058018.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020580/c02058019.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020580/c02058020.png" /> and the root vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020580/c02058021.png" /> can be chosen such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020580/c02058022.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020580/c02058023.png" />, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020580/c02058024.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020580/c02058025.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020580/c02058026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020580/c02058027.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020580/c02058028.png" />. There exists a normalization of the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020580/c02058029.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020580/c02058030.png" />, where the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020580/c02058031.png" /> obtained are rational. There exists a normalization of the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020580/c02058032.png" /> under which all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020580/c02058033.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020580/c02058034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020580/c02058035.png" /> and the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020580/c02058036.png" />, 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>