The group ${\mathrm{SL}}_2$ and the Lie algebra

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

Last updates: 21 May 2011

The group ${\mathrm{SL}}_2$ and the Lie algebra

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

${\mathrm{SL}}_2=\left\{\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\right|a,b,c,d\in 𝔽\text{with}ad-bc=1\}$

with product given by matrix multiplication. One parameter subgroups are

$x_{12}\left(t\right)=\left(\begin{array}{cc}1& t\\ 0& 1\end{array}\right),x_{21}\left(t\right)=\left(\begin{array}{cc}1& 0\\ t& 1\end{array}\right),h_{{\alpha }^{\vee }}\left(t\right)=\left(\begin{array}{cc}{e}^t& 0\\ 0& e^{-t}\end{array}\right)$,

and the Lie algebra

has basis $\{x,y,h\}$ where

$x=\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right),y=\left(\begin{array}{cc}0& 0\\ 1& 0\end{array}\right),h=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$.

Then is presented by generators $x,y,h$ with relations

$[x,y]=h,[h,x]=2x,[h,y]=-2y$.

The group ${\mathrm{SL}}_2$ is presented by generators

$x_{12}\left(t\right)=\left(\begin{array}{cc}1& t\\ 0& 1\end{array}\right),x_{-\alpha }\left(t\right)=\left(\begin{array}{cc}1& 0\\ t& 1\end{array}\right),t\in 𝔽,$

with relations

$x_{\pm \alpha }(s+t)=x_{\pm \alpha }\left(s\right)x_{\pm \alpha }\left(t\right),h_{{\alpha }^{\vee }}\left(c_1{c}_2\right)=h_{{\alpha }^{\vee }}\left(c_1\right)h_{{\alpha }^{\vee }}\left(c_2\right),$

and

$n_{\alpha }\left(t\right)x_{\alpha }\left(u\right)n_{\alpha }(-t)=x_{-\alpha }(-t^{-2}u)$,

where

$n_{\alpha }\left(t\right)=x_{\alpha }\left(t\right)x_{-\alpha }(-t^{-1})x_{\alpha }\left(t\right)\text{and}{h}_{{\alpha }^{\vee }}\left(t\right)=n_{\alpha }\left(t\right)n_{-\alpha }(-1).$

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.

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