Chevalley basis

{{inline |date=May 2024}}

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 $\backslash pm\backslash alpha\_i$. The Cartan-Weyl basis may be written as

:$[H\_i,H\_j]=0$

:$[H\_i,E\_\backslash alpha]=\backslash alpha\_i\; E\_\backslash alpha$

Defining the dual root or coroot of $\backslash alpha$ as

:$\backslash alpha^\backslash vee\; =\; \backslash frac\{2\backslash alpha\}\{(\backslash alpha,\backslash alpha)\}$

where $(\backslash cdot,\backslash cdot)$ is the euclidean inner product. One may perform a change of basis to define

:$H\_\{\backslash alpha\_i\}=(\backslash alpha\_i^\backslash vee,\; H)$

The Cartan integers are

:$A\_\{ij\}=(\backslash alpha\_i,\backslash alpha\_j^\backslash vee)$

The resulting relations among the generators are the following:

:$[H\_\{\backslash alpha\_i\},H\_\{\backslash alpha\_j\}]=0$

:$[H\_\{\backslash alpha\_i\},E\_\{\backslash alpha\_j\}]=A\_\{ji\}\; E\_\{\backslash alpha\_j\}$

:$[E\_\{-\backslash alpha\_i\},E\_\{\backslash alpha\_i\}]\; =\; H\_\{\backslash alpha\_i\}$

:$[E\_\{\backslash beta\},E\_\{\backslash gamma\}]=\backslash pm(p+1)E\_\{\backslash beta+\backslash gamma\}$

where in the last relation $p$ is the greatest positive integer such that $\backslash gamma\; -p\backslash beta$ is a root and we consider $E\_\{\backslash beta\; +\; \backslash gamma\}\; =\; 0$ if $\backslash beta\; +\; \backslash gamma$ is not a root.

For determining the sign in the last relation one fixes an ordering of roots which respects addition, i.e., if $\backslash beta\; \backslash prec\; \backslash gamma$ then $\backslash beta\; +\; \backslash alpha\; \backslash prec\; \backslash gamma\; +\; \backslash alpha$ provided that all four are roots. We then call $(\backslash beta,\; \backslash gamma)$ an extraspecial pair of roots if they are both positive and $\backslash beta$ is minimal among all $\backslash beta\_0$ that occur in pairs of positive roots $(\backslash beta\_0,\; \backslash gamma\_0)$ satisfying $\backslash beta\_0\; +\; \backslash gamma\_0\; =\; \backslash beta\; +\; \backslash gamma$. The sign in the last relation can be chosen arbitrarily whenever $(\backslash beta,\; \backslash gamma)$ is an extraspecial pair of roots. This then determines the signs for all remaining pairs of roots.