## The group ${\mathrm{SL}}_{3}$ and the Lie algebra ${\mathrm{𝔰𝔩}}_{3}$

Last update: 15 July 2012

## The group ${\mathrm{SL}}_{3}$

${W}_{0}=\left\{1,{s}_{1},{s}_{2},{s}_{1}{s}_{2},{s}_{2}{s}_{1},{s}_{1}{s}_{2}{s}_{2}\right\}$ contains 3 reflections, and ${s}_{1}{s}_{2}{s}_{1}={s}_{\theta },$ the reflections in respectively.

${\mathrm{SL}}_{3}$. Let $𝔤={\mathrm{𝔰𝔩}}_{3}$ be the span of the elements $Xα1 =E12, Xα2 =E23 X α1+α2 =E21, X-α1 =E21, X-α2 =E32, X-α1 -α2 =E31,$ $Hα1∨ =E11-E22 and Hα2∨ =E22-E33,$ where ${E}_{ij}$ is the $3×3$ matrix with a 1 in the $\left(i,j\right)$ entry and zeros elsewhere. Then $xα1(f) = ( 1 f 0 0 1 0 0 0 1 ) , xα1(f) = ( 1 0 0 0 1 f 0 0 1 ) and xα1 +α2(f) = ( 10f 010 001 ) .$ We compute $nα1(g) = ( 0g0 -g-100 001 ) and hα1∨(g) = ( g00 0-g-10 001 ),$ and $nα2(g) = ( 100 00g 0-g-10 ) and hα2∨(g) = ( 100 0g0 00-g-1 ).$ Then $G={\mathrm{SL}}_{3}\left(𝔽\right).$ The nontrivial commutator relation is $xα1(f1) xα2(f2) = xα2(f2) xα1(f1) xα1 +α2 (f1f2) ,$ and the center of $G$ is $Z(G) = { hα1∨ (g1) hα2∨ (g2) | g13=1 and g2 =g1-1 }.$

## Notes and References

These notes give the presentations of the Chevalley group ${\mathrm{SL}}_{3}$ and the Kac-Moody Lie algebra ${𝔰𝔩}_{3}$.

References?