= Solution
Here a <reduced root system> is crystallographic, as appropriate to a <semisimple Lie algebra>. It is a finite spanning set $\Phi\subset E\setminus\{0\}$ in a real <inner-product space> of dimension $r$, invariant under each <Weyl reflection>
$$
s_\alpha(x)=x-\frac{2(x,\alpha)}{(\alpha,\alpha)}\alpha,
$$
with $2(\beta,\alpha)/(\alpha,\alpha)\in\mathbb Z$ for every pair of roots and $\Phi\cap\mathbb R\alpha=\{\alpha,-\alpha\}$. A <base of a root system> $\Delta=(\alpha_1,\ldots,\alpha_r)$ is a <basis> such that the coefficients of every root are integers that are either all nonnegative or all nonpositive. These define the <positive system of a root system>. We use the existing <Cartan matrix> convention
$$
A_{ij}=\langle\alpha_i,\alpha_j^\vee\rangle=\frac{2(\alpha_i,\alpha_j)}{(\alpha_j,\alpha_j)}.
$$
Using the alternative denominator $(\alpha_i,\alpha_i)$ transposes all the matrices below.
The <classification of rank-two root systems> gives \b[$A_1\times A_1$, $A_2$, $B_2=C_2$, and $G_2$]. In orthonormal coordinates we may choose the following <simple roots> and <Cartan matrices>:
* For $A_1\times A_1$, take $\alpha_1=(1,0)$, $\alpha_2=(0,1)$ and $A=\begin{pmatrix}2&0\\0&2\end{pmatrix}$. The positive roots are $\alpha_1,\alpha_2$.
* For $A_2$, take $\alpha_1=(1,0)$, $\alpha_2=(-1/2,\sqrt3/2)$ and $A=\begin{pmatrix}2&-1\\-1&2\end{pmatrix}$. The positive roots are $\alpha_1,\alpha_2,\alpha_1+\alpha_2$.
* For $B_2$, take the long root $\alpha_1=(1,-1)$ and short root $\alpha_2=(0,1)$, giving $A=\begin{pmatrix}2&-2\\-1&2\end{pmatrix}$. The positive roots are $\alpha_1,\alpha_2,\alpha_1+\alpha_2,\alpha_1+2\alpha_2$. Interchanging the long and short convention gives type $C_2$, which is the same rank-two classification up to the usual identification.
* For $G_2$, take the short root $\alpha_1=(1,0)$ and long root $\alpha_2=(-3/2,\sqrt3/2)$, giving $A=\begin{pmatrix}2&-1\\-3&2\end{pmatrix}$. The positive roots are $\alpha_1,\alpha_2,\alpha_1+\alpha_2,2\alpha_1+\alpha_2,3\alpha_1+\alpha_2,3\alpha_1+2\alpha_2$.
Each complete <root system> consists of these positive roots and their negatives. To see why the list is exhaustive, the off-diagonal <Cartan matrix> entries of two distinct simple roots are nonpositive integers, and their product is $4\cos^2\theta<4$. Thus the product is $0,1,2$ or $3$, corresponding to simple-root angles $\pi/2,2\pi/3,3\pi/4$ or $5\pi/6$. These determine the four systems and their length ratios.
The <Weyl group of a rank-two root system> is generated by the two simple <Weyl reflections>. Their product is a rotation of order $m=2,3,4,6$, respectively, so the groups are the <dihedral groups> of order $2m=4,6,8,12$. The first is also the <Klein four-group> and the second the <symmetric group> $S_3$. The root-orthogonal lines cut the plane into $2m$ <Weyl chambers>, each of angle $\pi/m$. A chosen <fundamental chamber of a root system> is $(x,\alpha_1)>0$, $(x,\alpha_2)>0$; its walls are the two root-orthogonal lines. The <Weyl group> acts simply transitively on these open chambers.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2013/iii/paper-1-rank-two-root-systems.png]
{title=Roots, reflecting lines and a fundamental chamber for the four crystallographic rank-two root systems}
{height=700}
The crystallographic hypothesis matters: if the integer-pairing condition is dropped, reduced noncrystallographic systems of type $I_2(m)$ also occur. They are not additional Lie-algebra root systems.
Back to article page
