## The group ${\mathrm{SL}}_{2}$ and the Lie algebra ${\mathrm{𝔰𝔩}}_{2}$

Let $𝔽$ be a field (or a commutative ring) and let ${\mathrm{𝔤𝔩}}_{2}$ be the Lie algebra of $2×2$ matrices with entries in $𝔽$ and bracket given by $\left[p,q\right]=pq-qp$. The group

 ${\mathrm{SL}}_{2}=\left\{\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\phantom{\rule{0.5em}{0ex}}|\phantom{\rule{0.5em}{0ex}}a,b,c,d\in 𝔽\phantom{\rule{0.5em}{0ex}}\text{with}\phantom{\rule{0.5em}{0ex}}ad-bc=1\right\}$
with product given by matrix multiplication. One parameter subgroups are
 ${x}_{12}\left(t\right)=\left(\begin{array}{cc}1& t\\ 0& 1\end{array}\right),\phantom{\rule{2em}{0ex}}{x}_{21}\left(t\right)=\left(\begin{array}{cc}1& 0\\ t& 1\end{array}\right),\phantom{\rule{2em}{0ex}}{h}_{{\alpha }^{\vee }}\left(t\right)=\left(\begin{array}{cc}{e}^{t}& 0\\ 0& {e}^{-t}\end{array}\right)$,
and the Lie algebra
 ${\mathrm{𝔰𝔩}}_{2}=\left\{x\in {\mathrm{𝔤𝔩}}_{2}\phantom{\rule{0.5em}{0ex}}|\phantom{\rule{0.5em}{0ex}}\mathrm{tr}\left(x\right)=0\right\}$
has basis $\left\{x,y,h\right\}$ where
 $x=\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right),\phantom{\rule{2em}{0ex}}y=\left(\begin{array}{cc}0& 0\\ 1& 0\end{array}\right),\phantom{\rule{2em}{0ex}}h=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$.
Then ${\mathrm{𝔰𝔩}}_{2}$ is presented by generators $x,y,h$ with relations
 $\left[x,y\right]=h,\phantom{\rule{2em}{0ex}}\left[h,x\right]=2x,\phantom{\rule{2em}{0ex}}\left[h,y\right]=-2y$.
The group ${\mathrm{SL}}_{2}$ is presented by generators
 ${x}_{12}\left(t\right)=\left(\begin{array}{cc}1& t\\ 0& 1\end{array}\right),\phantom{\rule{2em}{0ex}}{x}_{-\alpha }\left(t\right)=\left(\begin{array}{cc}1& 0\\ t& 1\end{array}\right),\phantom{\rule{2em}{0ex}}t\in 𝔽,$
with relations
 ${x}_{±\alpha }\left(s+t\right)={x}_{±\alpha }\left(s\right){x}_{±\alpha }\left(t\right),\phantom{\rule{2em}{0ex}}{h}_{{\alpha }^{\vee }}\left({c}_{1}{c}_{2}\right)={h}_{{\alpha }^{\vee }}\left({c}_{1}\right){h}_{{\alpha }^{\vee }}\left({c}_{2}\right),\phantom{\rule{2em}{0ex}}$
and
 ${n}_{\alpha }\left(t\right){x}_{\alpha }\left(u\right){n}_{\alpha }\left(-t\right)={x}_{-\alpha }\left(-{t}^{-2}u\right)$,
where
 ${n}_{\alpha }\left(t\right)={x}_{\alpha }\left(t\right){x}_{-\alpha }\left(-{t}^{-1}\right){x}_{\alpha }\left(t\right)\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}{h}_{{\alpha }^{\vee }}\left(t\right)={n}_{\alpha }\left(t\right){n}_{-\alpha }\left(-1\right).$

## Notes and References

These notes follow Steinberg [St, ????].

## References

[St] R. Steinberg, Lectures on Chevalley groups, Notes prepared by John Faulkner and Robert Wilson, Yale University, New Haven, Conn., 1968. iii+277 pp. MR0466335.