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

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.

The first diagram is $D_4$, with central node ${\alpha }_2$. Its Cartan matrix is

The second diagram is $F_4$; the arrow points from the long root ${\alpha }_2$ toward the short root ${\alpha }_3$. Thus

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 $N=3$.

This is the G2 root system, with $\alpha$ long and $\beta$ short. Its positive roots areand the full root set includes their negatives. Thus $\dim \mathfrak{g}=12+2=14$. Finallyand simple roots have obtuse angle, so $\theta =5\pi /6$.

For $x\alpha +y\beta$, the simple-coroot coordinates are $2x-y$ and $-3x+2y$. Solving for the fundamental weights givesHence $C_1=2$, $C_2=3$, and $C_3=2$. The weight $\omega$ is the fundamental weight of the short root, and its highest-weight representation is the seven-dimensional fundamental representation of $G_2$. Its weights are zero and the six short roots.

Under $\mathfrak{sl}(2{)}_{\alpha }$, a root $x\alpha +y\beta$ has weight $2x-y$. Counting the twelve root spaces and the two-dimensional Cartan subalgebra gives multiplicities one at weights $\pm 2$, four at $\pm 1$, and four at zero. Therefore

Under $\mathfrak{sl}(2{)}_{\beta }$, the weight is $-3x+2y$. The multiplicities are two at $\pm 3$, one at $\pm 2$, two at $\pm 1$, and four at zero. HenceThe dimensions are respectively $3+4\cdot 2+3=14$ and $2\cdot 4+3+3=14$, verifying both direct-sum decompositions of the Adjoint representation.