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

A Weyl chamber is a connected component ofA root basis $\Delta \subseteq \Phi$ is a vector-space basis of $E$ such that every root is an integer combination of elements of $\Delta$ with all nonzero coefficients of one sign.

To construct one, choose a regular vector $\gamma \in E$, meaning $(\gamma ,\alpha )\ne 0$ for every root. DeclareThe positive roots in ${\Phi }_{\gamma }^{+}$ which cannot be written as sums of two positive roots form a root basis ${\Delta }_{\gamma }$. Vectors $\gamma$ in the same Weyl chamber give the same basis.

Write a positive nonsimple root asIf $(\alpha ,\beta )\le 0$ for every simple root with $n_{\beta }>0$, thenwhich is impossible. Hence $(\alpha ,\beta )>0$ for some simple $\beta$. The root-string property then gives $\alpha -\beta \in \Phi$, and its simple-root coefficients remain nonnegative. This is the simple-root subtraction lemma.

Induct on the height $\sum n_{\beta }$. Applying the induction hypothesis to $\alpha -\beta$ and appending $\beta$ writesso that every partial sum is a root.

Finally let $\alpha$ be simple and let $\gamma \ne \alpha$ be positive. In the simple-root expansion ofall coefficients except possibly that of $\alpha$ are unchanged, and at least one of those unchanged coefficients is positive. Since a root has coefficients all of one sign, the image cannot be negative. Thus $s_{\alpha }$ permutes ${\Phi }^{+}\setminus \{\alpha \}$, as stated by action of a simple reflection on positive roots.

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.