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 formula_1. The Cartan-Weyl basis may be written as
formula_2
formula_3
Defining the dual root or coroot of formula_4 as
formula_5
where formula_6 is the euclidean inner product. One may perform a change of basis to define
formula_7
The Cartan integers are
formula_8
The resulting relations among the generators are the following:
formula_9
formula_10
formula_11
formula_12
where in the last relation formula_13 is the greatest positive integer such that formula_14 is a root and we consider formula_15 if formula_16 is not a root.
For determining the sign in the last relation one fixes an ordering of roots which respects addition, i.e., if formula_17 then formula_18 provided that all four are roots. We then call formula_19 an extraspecial pair of roots if they are both positive and formula_20 is minimal among all formula_21 that occur in pairs of positive roots formula_22 satisfying formula_23. The sign in the last relation can be chosen arbitrarily whenever formula_19 is an extraspecial pair of roots. This then determines the signs for all remaining pairs of roots.