Cartan-Weyl basis
of a finite-dimensional semi-simple complex Lie algebra
A basis of consisting of elements of a Cartan subalgebra
of
and root vectors
,
, where
is the system of all non-zero roots of
with respect to
. The choice of a Cartan–Weyl basis is not unique. A root
,
, is identified, as a linear form on
, with the vector
such that
, where
is the Killing form in
. Here
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for each . If
, then
and the root vectors
can be chosen such that
. If
, then
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where . If
,
, then
. There exists a normalization of the vectors
for which
, where the numbers
obtained are rational. There exists a normalization of the vectors
under which all
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
,
and the numbers
, 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) |
Cartan-Weyl basis. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cartan-Weyl_basis&oldid=22255