The Lie algebra $G_2$

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 24 February 2012

7 dimensional representation of $G_2$

The complex simple Lie algebra of type $G_2$ is given by generators $e_1,e_2,h_1,h_2,f_1,f_2$ and relations $$\begin{array}{}\begin{array}{ccc}[e_1,f_2]=0,& [e_2,f_1]=0,& [h_1,h_2]=0,\end{array}\\ \begin{array}{cc}[e_1,f_1]=h_1,& [e_2,f_2]=h_2,\end{array}\\ \begin{array}{llll}[h_1,e_1]=2e_1,& [h_1,e_2]=-3e_2,& [h_1,f_1]=-2f_1,& [h_1,f_2]=3f_2,\\ [h_2,e_1]=-e_1,& [h_2,e_2]=2e_2,& [h_2,f_1]=f_1,& [h_2,f_2]=-2f_2,\end{array}\\ \begin{array}{ll}[e_1,\left[e_1\left[e_1[e_1,e_2]\right]\right]]=0,& [e_2,[e_2,e_1]]=0,\\ [f_1,\left[f_1\left[f_1[f_1,f_2]\right]\right]]=0,& [f_2,[f_2,f_1]]=0.\end{array}\end{array}$$ The Cartan matrix is $\left(\begin{array}{cc}2& -3\\ -1& 2\end{array}\right).$ The algebra $𝔤$ has basis $e_{{\alpha }_1},e_{{\alpha }_2},h_1,h_2,f_{{\alpha }_1},f_{{\alpha }_2}$ $$\begin{array}{lll}{e}_{{\alpha }_1}=e_1,& e_{{\alpha }_2}=e_2,& e_{{\alpha }_1+{\alpha }_2}=[e_1,e_2],\\ e_{2{\alpha }_1+{\alpha }_2}=\frac{1}{2}[e_1,e_{{\alpha }_1+{\alpha }_2}],& e_{3{\alpha }_1+{\alpha }_2}=\frac{1}{3}[e_1,e_{2{\alpha }_1+{\alpha }_2}],& e_{3{\alpha }_1+2{\alpha }_2}=[e_2,e_{3{\alpha }_1+{\alpha }_2}],\\ f_{{\alpha }_1}=f_1,& f_{{\alpha }_2}=f_2,& f_{{\alpha }_1+{\alpha }_2}=[f_1,f_2],\\ f_{2{\alpha }_1+{\alpha }_2}=\frac{1}{2}[f_1,f_{{\alpha }_1+{\alpha }_2}],& f_{3{\alpha }_1+{\alpha }_2}=\frac{1}{3}[f_1,f_{2{\alpha }_1+{\alpha }_2}],& f_{3{\alpha }_1+2{\alpha }_2}=[f_2,f_{3{\alpha }_1+{\alpha }_2}].\end{array}$$ A realization of the 7 dimensional representation of $𝔤$ is by the matrices $$\begin{array}{rcl}\phi \left(e_1\right)& =& \left(\begin{array}{ccccccc}0& 1& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& -2& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0& 0& 0\end{array}\right),\\ \phi \left(e_2\right)& =& \left(\begin{array}{ccccccc}0& 0& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\end{array}\right),\\ \phi \left(f_1\right)& =& \left(\begin{array}{ccccccc}0& 0& 0& 0& 0& 0& 0\\ 1& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& -1& 0& 0& 0& 0\\ 0& 0& 0& 2& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 1& 0\end{array}\right),\\ \phi \left(f_2\right)& =& \left(\begin{array}{ccccccc}0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\end{array}\right),\\ \phi \left(h_1\right)& =& \left(\begin{array}{ccccccc}1& 0& 0& 0& 0& 0& 0\\ 0& -1& 0& 0& 0& 0& 0\\ 0& 0& 2& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& -2& 0& 0\\ 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& -1\end{array}\right),and\\ \phi \left(h_2\right)& =& \left(\begin{array}{ccccccc}0& 0& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0& 0\\ 0& 0& -1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& -1& 0\\ 0& 0& 0& 0& 0& 0& 0\end{array}\right).\end{array}$$ Then $$\begin{array}{rcl}\phi \left(e_{{\alpha }_1+{\alpha }_2}\right)& =& \left(\begin{array}{ccccccc}0& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 2& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& -1\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\end{array}\right),\\ \phi \left(e_{2{\alpha }_1+{\alpha }_2}\right)& =& \left(\begin{array}{ccccccc}0& 0& 0& 2& 0& 0& 0\\ 0& 0& 0& 0& -1& 0& 0\\ 0& 0& 0& 0& 0& -1& 0\\ 0& 0& 0& 0& 0& 0& -1\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\end{array}\right),\\ \phi \left(e_{3{\alpha }_1+{\alpha }_2}\right)& =& \left(\begin{array}{ccccccc}0& 0& 0& 0& -1& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\end{array}\right),and\\ \phi \left(e_{3{\alpha }_1+2{\alpha }_2}\right)& =& \left(\begin{array}{ccccccc}0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\end{array}\right).\end{array}$$ Also, $$\begin{array}{rcl}\phi \left(f_{{\alpha }_1+{\alpha }_2}\right)& =& \left(\begin{array}{ccccccc}0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ -1& 0& 0& 0& 0& 0& 0\\ 0& -1& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& -2& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0\end{array}\right),\\ \phi \left(f_{2{\alpha }_1+{\alpha }_2}\right)& =& \left(\begin{array}{ccccccc}0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 1& 0& 0& 0& 0& 0& 0\\ 0& -1& 0& 0& 0& 0& 0\\ 0& 0& -1& 0& 0& 0& 0\\ 0& 0& 0& -2& 0& 0& 0\end{array}\right),\\ \phi \left(f_{3{\alpha }_1+{\alpha }_2}\right)& =& \left(\begin{array}{ccccccc}0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 1& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& -1& 0& 0& 0& 0\end{array}\right),and\\ \phi \left(f_{3{\alpha }_1+2{\alpha }_2}\right)& =& \left(\begin{array}{ccccccc}0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 1& 0& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0& 0\end{array}\right).\end{array}$$

Let $G$ be the corresponding Lie group of type $G_2,$ the group generated by all the $x_{\alpha }\left(c\right)=\exp c{e}_{\alpha }$ for $\alpha \in R$ and $c\in ℂ.$ Define $$n_i\left(c\right)=x_{{\alpha }_i}\left(c\right)x_{-{\alpha }_i}(-c^{-1})x_{{\alpha }_i}\left(c\right),$$ and $$h_{{\alpha }_i^{\vee }}\left(c\right)=n_i\left(c\right)n_i{\left(1\right)}^{-1}.$$ We use the exponentiated matrices to compute the following relations in $G.$ (These relations are described in [St3].)

The first relations are commutator relations between unipotent elements. $$\begin{array}{rcl}{x}_{{\alpha }_1}\left(c\right)x_{{\alpha }_2}\left(d\right)& =& x_{{\alpha }_2}\left(d\right)x_{{\alpha }_1}\left(c\right)x_{{\alpha }_1+{\alpha }_2}\left(cd\right)x_{2{\alpha }_1+{\alpha }_2}(-c^{2}d)x_{3{\alpha }_1+{\alpha }_2}\left(c^{3}d\right)x_{3{\alpha }_1+2{\alpha }_2}(-2c^3{d}^2)\\ x_{{\alpha }_1}\left(c\right)x_{{\alpha }_1+{\alpha }_2}\left(d\right)& =& x_{{\alpha }_1+{\alpha }_2}\left(d\right)x_{{\alpha }_1}\left(c\right)x_{2{\alpha }_1+{\alpha }_2}\left(2cd\right)x_{3{\alpha }_1+{\alpha }_2}(-3c^{2}d)x_{3{\alpha }_1+2{\alpha }_2}\left(3c{d}^2\right)\\ x_{{\alpha }_1}\left(c\right)x_{2{\alpha }_1+{\alpha }_2}\left(d\right)& =& x_{2{\alpha }_1+{\alpha }_2}\left(d\right)x_{{\alpha }_1}\left(c\right)x_{3{\alpha }_1+{\alpha }_2}\left(3c{d}^2\right)\\ x_{{\alpha }_1}\left(c\right)x_{3{\alpha }_1+{\alpha }_2}\left(d\right)& =& x_{3{\alpha }_1+{\alpha }_2}\left(d\right)x_{{\alpha }_1}\left(c\right)\\ x_{{\alpha }_1}\left(c\right)x_{3{\alpha }_1+2{\alpha }_2}\left(d\right)& =& x_{3{\alpha }_1+2{\alpha }_2}\left(d\right)x_{{\alpha }_1}\left(c\right)\\ x_{{\alpha }_2}\left(c\right)x_{{\alpha }_1+{\alpha }_2}\left(d\right)& =& x_{{\alpha }_1+{\alpha }_2}\left(d\right)x_{{\alpha }_2}\left(c\right)\\ x_{{\alpha }_2}\left(c\right)x_{2{\alpha }_1+{\alpha }_2}\left(d\right)& =& x_{2{\alpha }_1+{\alpha }_2}\left(d\right)x_{{\alpha }_2}\left(c\right)\\ x_{{\alpha }_2}\left(c\right)x_{3{\alpha }_1+{\alpha }_2}\left(d\right)& =& x_{3{\alpha }_1+{\alpha }_2}\left(d\right)x_{{\alpha }_2}\left(c\right)x_{3{\alpha }_1+2{\alpha }_2}\left(cd\right)\\ x_{{\alpha }_2}\left(c\right)x_{3{\alpha }_1+2{\alpha }_2}\left(d\right)& =& x_{3{\alpha }_1+2{\alpha }_2}\left(d\right)x_{{\alpha }_2}\left(c\right)\\ x_{{\alpha }_1+{\alpha }_2}\left(c\right)x_{2{\alpha }_1+{\alpha }_2}\left(d\right)& =& x_{2{\alpha }_1+{\alpha }_2}\left(d\right)x_{{\alpha }_1+{\alpha }_2}\left(c\right)x_{3{\alpha }_1+2{\alpha }_2}(-3cd)\\ x_{{\alpha }_1+{\alpha }_2}\left(c\right)x_{3{\alpha }_1+{\alpha }_2}\left(d\right)& =& x_{3{\alpha }_1+{\alpha }_2}\left(d\right)x_{{\alpha }_1+{\alpha }_2}\left(c\right)\\ x_{{\alpha }_1+{\alpha }_2}\left(c\right)x_{3{\alpha }_1+2{\alpha }_2}\left(d\right)& =& x_{3{\alpha }_1+2{\alpha }_2}\left(d\right)x_{{\alpha }_1+{\alpha }_2}\left(c\right)\\ x_{2{\alpha }_1+{\alpha }_2}\left(c\right)x_{3{\alpha }_1+{\alpha }_2}\left(d\right)& =& x_{3{\alpha }_1+{\alpha }_2}\left(d\right)x_{2{\alpha }_1+{\alpha }_2}\left(c\right)\\ x_{2{\alpha }_1+{\alpha }_2}\left(c\right)x_{3{\alpha }_1+2{\alpha }_2}\left(d\right)& =& x_{3{\alpha }_1+2{\alpha }_2}\left(d\right)x_{2{\alpha }_1+{\alpha }_2}\left(c\right)\\ x_{3{\alpha }_1+{\alpha }_2}\left(c\right)x_{3{\alpha }_1+2{\alpha }_2}\left(d\right)& =& x_{3{\alpha }_1+2{\alpha }_2}\left(d\right)x_{3{\alpha }_1+{\alpha }_2}\left(c\right)\end{array}$$
The other needed relations describe how the Weyl group acts on the unipotent elements of $G.$ $$\begin{array}{rcl}{n}_1\left(a\right)e_1{n_1\left(a\right)}^{-1}& =& -a^{-2}{f}_1\\ n_1\left(a\right)e_2{n_1\left(a\right)}^{-1}& =& a^3{e}_{3{\alpha }_1+{\alpha }_2}\\ n_1\left(a\right)e_{{\alpha }_1+{\alpha }_2}{n_1\left(a\right)}^{-1}& =& -a{e}_{2{\alpha }_1+{\alpha }_2}\\ n_1\left(a\right)e_{2{\alpha }_1+{\alpha }_2}{n_1\left(a\right)}^{-1}& =& a^{-1}{e}_{{\alpha }_1+{\alpha }_2}\\ n_1\left(a\right)e_{3{\alpha }_1+{\alpha }_2}{n_1\left(a\right)}^{-1}& =& -a^{-3}{e}_{{\alpha }_2}\\ n_1\left(a\right)e_{3{\alpha }_1+2{\alpha }_2}{n_1\left(a\right)}^{-1}& =& e_{3{\alpha }_1+2{\alpha }_2}\\ n_2\left(b\right)e_2{n_2\left(b\right)}^{-1}& =& -b^{-2}{f}_2\\ n_2\left(b\right)e_1{n_2\left(b\right)}^{-1}& =& -b{e}_{{\alpha }_1+{\alpha }_2}\\ n_2\left(b\right)e_{{\alpha }_1+{\alpha }_2}{n_2\left(b\right)}^{-1}& =& b^{-1}{e}_{{\alpha }_1}\\ n_2\left(b\right)e_{2{\alpha }_1+{\alpha }_2}{n_2\left(b\right)}^{-1}& =& e_{2{\alpha }_1+{\alpha }_2}\\ n_2\left(b\right)e_{3{\alpha }_1+{\alpha }_2}{n_2\left(b\right)}^{-1}& =& b{e}_{3{\alpha }_1+2{\alpha }_2}\\ n_2\left(b\right)e_{3{\alpha }_1+2{\alpha }_2}{n_2\left(b\right)}^{-1}& =& -b^{-1}{e}_{3{\alpha }_1+{\alpha }_2}\end{array}$$

Notes and References

Where are these from?

References

References?

page history