Choose simple roots $\alpha,\beta$. Their inner product is nonpositive, so their angle lies in $[\pi/2,\pi)$. Part i leaves four possibilities for $mn$:

* $mn=0$: angle $\pi/2$, giving $A_1\times A_1$.
* $mn=1$: angle $2\pi/3$ and equal root lengths, giving $A_2$.
* $mn=2$: angle $3\pi/4$ and squared-length ratio $2$, giving $B_2$.
* $mn=3$: angle $5\pi/6$ and squared-length ratio $3$, giving $G_2$.

The root strings generated by the two simple reflections produce exactly the roots in those four standard systems. Hence these are all possibilities, proving the <classification of rank-two root systems>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 102 / 4 / ii / Solution

The <classification of rank-two root systems> gives $A_1\times A_1$, $A_2$, $B_2$, and $G_2$. The first is reducible and corresponds to a semisimple but nonsimple algebra. Hence the complex simple rank-two algebras are $A_2$, $B_2=C_2$, and $G_2$, so $N=3$.

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 302 / 4 / c / Solution

Every reduced crystallographic rank-two root system is isomorphic to $A_1\times A_1$, $A_2$, $B_2$, or $G_2$. Their simple-root angles are respectively $\pi/2$, $2\pi/3$, $3\pi/4$, and $5\pi/6$.


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 Incoming links: Classification of rank-two root systems