In mathematics, a Chevalley basis for a simple complex Lie algebra is a basis constructed by Claude Chevalley with the property that all structure constants are integers.
Chevalley used these bases to construct analogues of Lie groups over finite fields, called Chevalley groups.
The Chevalley basis is the Cartan-Weyl basis, but with a different normalization.
The generators of a Lie group are split into the generators H and E indexed by simple roots and their negatives
The Cartan-Weyl basis may be written as Defining the dual root or coroot of
One may perform a change of basis to define The Cartan integers are The resulting relations among the generators are the following: where in the last relation
is the greatest positive integer such that
For determining the sign in the last relation one fixes an ordering of roots which respects addition, i.e., if
an extraspecial pair of roots if they are both positive and
that occur in pairs of positive roots
The sign in the last relation can be chosen arbitrarily whenever
is an extraspecial pair of roots.
This then determines the signs for all remaining pairs of roots.
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