The group ${\mathrm{SL}}_3$ and the Lie algebra ${\mathrm{𝔰𝔩}}_3$

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

Last update: 15 July 2012

The group ${\mathrm{SL}}_3$

$W_0=\{1,s_1,s_2,s_1{s}_2,s_2{s}_1,s_1{s}_2{s}_2\}$ contains 3 reflections, $s_1, s_2$ and $s_1{s}_2{s}_1=s_{\theta },$ the reflections in ${𝔥}^{{\alpha }_1^{\vee }}, {𝔥}^{{\alpha }_2^{\vee }}, {𝔥}^{{\theta }^{\vee }},$ respectively.

${\mathrm{SL}}_3$. Let $𝔤={\mathrm{𝔰𝔩}}_3$ be the span of the elements $$X_{{\alpha }_1}=E_{12},X_{{\alpha }_2}=E_{23}{X}_{{\alpha }_1+{\alpha }_2}=E_{21},X_{-{\alpha }_1}=E_{21},X_{-{\alpha }_2}=E_{32},X_{-{\alpha }_1-{\alpha }_2}=E_{31},$$ $$H_{{\alpha }_1^{\vee }}=E_{11}-E_{22}\text{and}{H}_{{\alpha }_2^{\vee }}=E_{22}-E_{33},$$ where $E_{ij}$ is the $3\times 3$ matrix with a 1 in the $(i,j)$ entry and zeros elsewhere. Then $$x_{{\alpha }_1}\left(f\right)=\left(\begin{array}{ccc}1& f& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right),x_{{\alpha }_1}\left(f\right)=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& f\\ 0& 0& 1\end{array}\right)\text{and}{x}_{{\alpha }_1+{\alpha }_2}\left(f\right)=\left(\begin{array}{ccc}1& 0& f\\ 0& 1& 0\\ 0& 0& 1\end{array}\right).$$ We compute $$n_{{\alpha }_1}\left(g\right)=\left(\begin{array}{ccc}0& g& 0\\ -g^{-1}& 0& 0\\ 0& 0& 1\end{array}\right)\text{and}{h}_{{\alpha }_1^{\vee }}\left(g\right)=\left(\begin{array}{ccc}g& 0& 0\\ 0& -g^{-1}& 0\\ 0& 0& 1\end{array}\right),$$ and $$n_{{\alpha }_2}\left(g\right)=\left(\begin{array}{ccc}1& 0& 0\\ 0& 0& g\\ 0& -g^{-1}& 0\end{array}\right)\text{and}{h}_{{\alpha }_2^{\vee }}\left(g\right)=\left(\begin{array}{ccc}1& 0& 0\\ 0& g& 0\\ 0& 0& -g^{-1}\end{array}\right).$$ Then $G={\mathrm{SL}}_3\left(𝔽\right).$ The nontrivial commutator relation is $$x_{{\alpha }_1}\left(f_1\right)x_{{\alpha }_2}\left(f_2\right)=x_{{\alpha }_2}\left(f_2\right)x_{{\alpha }_1}\left(f_1\right)x_{{\alpha }_1+{\alpha }_2}\left(f_1{f}_2\right),$$ and the center of $G$ is $$Z\left(G\right)=\left\{h_{{\alpha }_1^{\vee }}\left(g_1\right)h_{{\alpha }_2^{\vee }}\left(g_2\right)\right|g_1^3=1\text{and}{g}_2=g_1^{-1}\}.$$

Notes and References

These notes give the presentations of the Chevalley group ${\mathrm{SL}}_3$ and the Kac-Moody Lie algebra ${𝔰𝔩}_3$.

References

References?

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