Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 102 / 3 / b

The diagonal Cartan subalgebra consists ofLet ${\epsilon }_i\in {\mathfrak{t}}^{\ast }$ read off $t_i$. The roots arethe D3 root system.

One root basis isAll roots have the same length. The nonzero inner products are $({\alpha }_1,{\alpha }_2)=({\alpha }_1,{\alpha }_3)=-1$, so the labeled Dynkin diagram isIt is the three-node $A_3$ diagram.

The simple reflections act on the root basis byby the Weyl reflection formula $s_{\alpha }(\beta )=\beta -⟨\beta ,{\alpha }^{\vee }⟩\alpha$.

The linear mappreserves $\Phi$ and exchanges ${\alpha }_2$ with ${\alpha }_3$ while fixing ${\alpha }_1$. It is not in the $D_3$ Weyl group, whose signed permutations change an even number of signs. Thus it is an outer automorphism of the root system.

The preceding Dynkin diagram identifies the root system $D_3$ with $A_3$. The classification of finite-dimensional complex Simple Lie algebras by connected Dynkin diagrams therefore givesThis is the Isomorphism between so6 and sl4. Both algebras have dimension $15$, consistently with the classification.