# Sign conventions for a Chevalley basis of a simple complex Lie algebra

Let $R$ be the root system of a simple complex Lie algebra $g$ with respect to some Cartan subalgebra $h$. Chevalley showed there is a basis of $g$ given by the simple coroots {$H_{\alpha_i}=\alpha_i^\vee\in h$} and root vectors $X_\alpha\in g_\alpha$ for each $\alpha\in R$. This basis has the following properties:

$[H_{\alpha_i},H_{\alpha_j}]=0$

$[H_{\alpha_i},X_\beta]=\beta(H_{\alpha_i})X_\beta$

$[X_{\alpha},X_{-\alpha}]=H_\alpha=\alpha^\vee\in h$

($\ast$) $[X_\alpha,X_\beta]=\pm(p+1)X_{\alpha+\beta}$, when $\alpha+\beta\in R$ and $p$ is the greatest positive integer such that $\beta-p\alpha\in R$. Otherwise, if $\alpha+\beta$ is not a root, then the bracket is zero.

References for this can be found in Serre's book on semisimple complex Lie algebras or Humphrey's book or Wikipedia.

Does anybody know a simple way to determine the sign $\pm$ in the fourth property ($\ast$)?

I cannot find a reference and my French is not good, so reading the original works by Chevalley and Tits isn't a viable option. In particular, I need to find a sign convention that will work for $g$ of type $F_4$.

Thanks so much.

• The Chevalley basis is only unique up to automorphisms of the Lie algebra and sign changes of the $X_\alpha$. (You can replace $(X_\alpha, X_{-\alpha})$ above by $(-X_\alpha, -X_{-\alpha})$ for any root $\alpha$ and all the axioms remain true.) I'm not sure whether a canonical choice is possible. However, it's possible that someone wrote down a possible choice for $F_4$ somewhere... (I suppose you can fix $X_\alpha$ for $\alpha$ simple, define the other positive ones using fixed instances of () with your favourite signs, and then work out the rest of () with a lot of patience.) Dec 17 '11 at 23:16
• How can the knowledge of these signs be useful? Dec 17 '11 at 23:19
• I'm working on a calculation in the exterior algebra $\bigwedge(g\oplus g)$ where $g$ is type $F_4$. I was hoping to do a small part of the calculation via a computer program, but getting a working model of $g$ is the first step. Dec 17 '11 at 23:33
• I've also had this issue come up when trying to do by-hand computations; in my experience, the best thing to do is to find a computer algebra package that implements some choice of these signs and then just let the computer handle it. I use Sage with a MAGMA interface and it's worked quite nicely, since MAGMA has all of this Lie-theoretic information built in already. (Feel free to drop me an email if you want to know more). Dec 19 '11 at 16:45

There is a good discussion of these issues in the paper of A. Cohen, S. Murray and D.E. Taylor, "Computing in groups of Lie type", Math. Comp. 73, Number 247, 1477–1498, (2003), especially section 3 (referring to earlier work, e.g., of Carter). They explain in particular how the signs can be all reduced to so-called "extraspecial pairs", which can be chosen arbitrarily.

In Magma at least, one can see which extraspecial signs have been chosen using the "ExtraspecialSigns" command. For instance, one can see using this that GAP and Magma use (or used, I haven't checked the latest versions...) different constants for E_8.

• Cohen has long been active (partly with de Graaf) in such Lie-theoretic computational problems, though I don't know the status of software from his LiE project. By the way, the paper cited here appeared in 2004 and is freely available online: ams.org/journals/mcom/2004-73-247/S0025-5718-03-01582-5/… Dec 18 '11 at 15:45

As Florian suggests, there is no canonical choice of structure constants in Chevalley's approach (or any other I'm aware of). But for the irreducible root systems, especially those of exceptional type, specific sign choices have been made in various papers (probably with some duplication of effort). One explicit source for type $F_4$ is Table 1 in an old paper by Toshiaki Shoji, published in J. Fac. Sci. Univ. Tokyo Sect. IA Math. 21 (1974), 1-17. This paper deals with conjugacy classes of Chevalley groups of type $F_4$ over finite fields of odd characteristic. Though I've never checked the arithmetic, Shoji's papers are usually reliable.

Over the years I've encountered explicit tables for other root systems but don't have these at hand. Computer methods have been used by N.A. Vavilov for root systems of type $E$ in one paper (where he notes that signs for $F_4$ can be deduced from those for $E_6$, via folding of the Dynkin diagam). Here is the full MathSciNet citation:

MR1875718 (2002k:17022) Vavilov, N. A. Do it yourself structure constants for Lie algebras of types $E_l$. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 281 (2001), Vopr. Teor. Predst. Algebr. i Grupp. 8, 60–104, 281; translation in J. Math. Sci. (N. Y.) 120 (2004), no. 4, 1513–1548.

Probably the recent computational work of W.A. de Graaf is relevant too.

• Thank you very much for this reference. In particular, because I'm already working with the $F_4\rightarrow E_6$ embedding in a related project and I'm very familiar with Mathematica. THANKS! :D Dec 18 '11 at 2:42