Skip to main content
1 of 4
D_S
  • 6.2k
  • 1
  • 15
  • 42

Reference Request: Structure constants for G2

Let $G$ be a split semisimple real Lie group in characteristic zero, and let $B=TU$ be the unipotent radical of a Borel subgroup. Fix an ordering on the roots $\Phi^+$ of $T$ in $U$, and for each root subgroup $U_{\alpha}$ of $U$, let $x_{\alpha}: \mathbb R \rightarrow U_{\alpha}$ be an isomorphism.

For all $\alpha, \beta \in \Phi^+$, there exist unique real numbers $C_{\alpha,\beta,i,j}$ such that for all $x, y \in \mathbb R$,

$$u_{\alpha}(x) u_{\beta}(y) u_{\alpha}(x)^{-1} = u_{\beta}(y) \prod\limits_{\substack{i,j>0\\ i\alpha + j \beta \in \Phi^+}} x_{i\alpha+j\beta}(C_{\alpha,\beta,i,j}x^iy^j)$$

I want to work out some examples with unipotent groups of exceptional semisimple groups, and am looking for table of structure constants for the root system G2. Does anyone know a reference where an ordering on the roots is chosen and these constants are written down?

D_S
  • 6.2k
  • 1
  • 15
  • 42