## The Lie algebra ${G}_{2}$

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 $[e1,f2]=0, [e2,f1]=0, [h1,h2]=0, [e1,f1]=h1, [e2,f2]=h2, [h1,e1]=2e1, [h1,e2]=-3e2, [h1,f1]=-2f1, [h1,f2]=3f2, [h2,e1]=-e1, [h2,e2]=2e2, [h2,f1]=f1, [h2,f2]=-2f2, [e1,[e1[e1[e1,e2]]]]=0, [e2,[e2,e1]]=0, [f1,[f1[f1[f1,f2]]]]=0, [f2,[f2,f1]]=0.$ The Cartan matrix is $\left(\begin{array}{cc}2& -3\\ -1& 2\end{array}\right).$ The algebra $𝔤$ has basis ${e}_{{\alpha }_{1}},{e}_{{\alpha }_{2}},{h}_{1},{h}_{2},{f}_{{\alpha }_{1}},{f}_{{\alpha }_{2}}$ $eα1=e1, eα2=e2, eα1+α2=[e1,e2], e2α1+α2= 1 2 [e1,eα1+α2], e3α1+α2= 1 3 [e1,e2α1+α2], e3α1+2α2=[e2,e3α1+α2], fα1=f1, fα2=f2, fα1+α2=[f1,f2], f2α1+α2= 1 2 [f1,fα1+α2], f3α1+α2= 1 3 [f1,f2α1+α2], f3α1+2α2=[f2,f3α1+α2].$ A realization of the 7 dimensional representation of $𝔤$ is by the matrices $φ(e1) = 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 , φ(e2) = 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , φ(f1) = 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 , φ(f2) = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 , φ(h1) = 1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 -1 ,and φ(h2) = 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 .$ Then $φ(eα1+α2) = 0 0 1 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , φ(e2α1+α2) = 0 0 0 2 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , φ(e3α1+α2) = 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ,and φ(e3α1+2α2) = 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 .$ Also, $φ(fα1+α2) = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2 0 0 0 0 0 0 0 1 0 0 , φ(f2α1+α2) = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -2 0 0 0 , φ(f3α1+α2) = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 ,and φ(f3α1+2α2) = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 .$

Let $G$ be the corresponding Lie group of type ${G}_{2},$ the group generated by all the ${x}_{\alpha }\left(c\right)=\mathrm{exp}c{e}_{\alpha }$ for $\alpha \in R$ and $c\in ℂ.$ Define $ni(c) = xαi(c) x-αi(-c-1) xαi(c),$ and $hαi∨(c) = ni(c) ni(1)-1.$ 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. $xα1(c) xα2(d) = xα2(d) xα1(c) xα1+α2(cd) x2α1+α2(-c2d) x3α1+α2(c3d) x3α1+2α2(-2c3d2) xα1(c) xα1+α2(d) = xα1+α2(d) xα1(c) x2α1+α2(2cd) x3α1+α2(-3c2d) x3α1+2α2(3cd2) xα1(c) x2α1+α2(d) = x2α1+α2(d) xα1(c) x3α1+α2(3cd2) xα1(c) x3α1+α2(d) = x3α1+α2(d) xα1(c) xα1(c) x3α1+2α2(d) = x3α1+2α2(d) xα1(c) xα2(c) xα1+α2(d) = xα1+α2(d) xα2(c) xα2(c) x2α1+α2(d) = x2α1+α2(d) xα2(c) xα2(c) x3α1+α2(d) = x3α1+α2(d) xα2(c) x3α1+2α2(cd) xα2(c) x3α1+2α2(d) = x3α1+2α2(d) xα2(c) xα1+α2(c) x2α1+α2(d) = x2α1+α2(d) xα1+α2(c) x3α1+2α2(-3cd) xα1+α2(c) x3α1+α2(d) = x3α1+α2(d) xα1+α2(c) xα1+α2(c) x3α1+2α2(d) = x3α1+2α2(d) xα1+α2(c) x2α1+α2(c) x3α1+α2(d) = x3α1+α2(d) x2α1+α2(c) x2α1+α2(c) x3α1+2α2(d) = x3α1+2α2(d) x2α1+α2(c) x3α1+α2(c) x3α1+2α2(d) = x3α1+2α2(d) x3α1+α2(c)$
The other needed relations describe how the Weyl group acts on the unipotent elements of $G.$ $n1(a) e1 n1(a)-1 = -a-2 f1 n1(a) e2 n1(a)-1 = a3e3α1+α2 n1(a) eα1+α2 n1(a)-1 = -ae2α1+α2 n1(a) e2α1+α2 n1(a)-1 = a-1 eα1+α2 n1(a) e3α1+α2 n1(a)-1 = -a-3 eα2 n1(a) e3α1+2α2 n1(a)-1 = e3α1+2α2 n2(b) e2 n2(b)-1 = -b-2 f2 n2(b) e1 n2(b)-1 = -beα1+α2 n2(b) eα1+α2 n2(b)-1 = b-1 eα1 n2(b) e2α1+α2 n2(b)-1 = e2α1+α2 n2(b) e3α1+α2 n2(b)-1 = be3α1+2α2 n2(b) e3α1+2α2 n2(b)-1 = -b-1 e3α1+α2$

## Notes and References

Where are these from?

References?