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

A Cartan subalgebra of a complex semisimple Lie algebra is a maximal abelian subalgebra consisting of semisimple elements.

For a Cartan subalgebra $\mathfrak{h}$, the root set consists of the nonzero linear functionals $\alpha \in {\mathfrak{h}}^{\ast }$ for which ${\mathfrak{g}}_{\alpha }=\{X:[H,X]=\alpha (H)X\}$ is nonzero. It is the root system of the Lie algebra.

A root string through $\beta$ in the $\alpha$ direction is the uninterrupted sequence $\beta -p\alpha ,\dots ,\beta +q\alpha$ of roots, with $p-q=⟨\beta ,{\alpha }^{\vee }⟩$.

For ordered simple roots, the Cartan matrix is $A_{ij}=⟨{\alpha }_i,{\alpha }_j^{\vee }⟩=2({\alpha }_i,{\alpha }_j)/({\alpha }_j,{\alpha }_j)$. It determines their relative lengths and angles.