User brandyn - MathOverflow

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

2011-12-17T22:57:24Z

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.

Answer by brandyn for What role does the "dual Coxeter number" play in Lie theory (and should it be called the "Kac number")?

2011-12-18T04:27:27Z

The dual Coxeter number $h$ comes up in a conjecture of Cachazo-Douglas-Seilberg-Witten which was motivated by supersymmetric gauge theory. Let $R:=\bigwedge( g\oplus g) = \bigwedge g\otimes\bigwedge g$, where $g$ is a finite dimensional simple complex Lie algebra. Let {$e_i$} be some basis of $g$ and let {$f_i$} denote the dual basis with respect to the normalized Killing form. Consider three different embeddings of $g$ into the 2-graded part of $R$ (which are independent of our chosen basis):

$C_1=${$\sum_i [x,e_i]\wedge f_i \otimes 1: x\in g$}$\subset \bigwedge^2 g\otimes \bigwedge^0 g$

$C_2=${$1\otimes \sum_i [x,e_i]\wedge f_i : x\in g$}$\subset \bigwedge^0 g\otimes \bigwedge^2 g$

$C_3=${$\sum_i [x,e_i]\otimes f_i : x\in g$}$\subset \bigwedge^1 g\otimes \bigwedge^1 g$

Let $J$ be the ideal of $R$ generated by $C_1,C_2,C_3$ and let $A$ denote the $g$-algebra $A:=R/J$. Lastly, let $$S=\sum_i e_i\otimes f_i\in \bigwedge^1g\otimes\bigwedge^1g,$$ which also does not depend on the choice of basis.

The CDSW conjecture is:

The subalgebra $A^g$ of $g$-invariants in $A$ is generated as an algebra by the element $S$. Furthermore, $S^h=0$ and $S^{h-1}\neq 0$. Thus, $$A^g\simeq \mathbb{C}[S]/\langle S^h\rangle.$$

I know this isn't an answer to your question, but it is another interesting example of where the dual Coxeter number makes the numerology work. The conjecture is open for type $F_4$ and $E_6,E_7,E_8$, but settled in the other cases. Also, I recently asked a question on MathOverFlow related to this topic and Jim helped me out on it considerably.

Lastly, for a reference, see On the Cachazo-Douglas-Seiberg-Witten conjecture for simple Lie algebras paper by Shrawan Kumar, [J. Amer. Math. Soc. 21 (2008), no. 3, 797--808; MR2393427 (2009e:17013)].

Multiplication tables for H*(G/P)?

2011-02-09T20:59:08Z

Hi everyone. My recent work has me developing software to compute in $H^\ast(G/P)$, where $G$ is a complex connected semisimple algebraic group and $P$ is a standard parabolic subgroup (usually, $B$ or a maximal $P$). While my programs are built on sound theory, one can never be too sure. It's always good practice to check your work.

I'm looking for references of multiplication tables of these cohomology rings.

In particular, I'm interested in cases where $G$ is NOT simply-laced (that is, Lie type $B_n$, $C_n$, $F_4$, $G_2$), though simply-laced tables would be nice too. Any tables would depend on a choice of additive basis for $H^\ast(G/P)$. I typically use cohomology classes either Hom-dual or Poincare dual to the usual Schubert varieties living in $G/P$, and I like to parameterize my Schubert varieties with $W^P$, the minimal length coset representatives of $W/W_P$ where $W$ is the Weyl group and $W_P$ is the Weyl group of the Levi associated to $P$. Tables using this convention would be great. Of course, tables in any basis would be fine. :D

Thanks so much.

Comment by brandyn

2011-12-18T02:42:50Z

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

Comment by brandyn

2011-12-17T23:33:28Z

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.

Comment by brandyn

2011-02-11T15:47:20Z

Actually, I was using Haibao's result's for my program (which I've written in Mathematica).