Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 102 / 2

The Cartan solvability criterion states that a finite-dimensional complex Lie algebra $\mathfrak{g}$ is solvable exactly whenwhere $\kappa$ is the Killing form. In the matrix form of the criterion, a Lie subalgebra $\mathfrak{g}\subseteq \mathfrak{gl}(V)$ is solvable iffor all $x\in [\mathfrak{g},\mathfrak{g}]$ and $y\in \mathfrak{g}$.

A torus $\mathfrak{t}$ in a Lie algebra is an abelian subalgebra whose elements act semisimply in the Adjoint representation. Its weight-space decomposition is

Invariance of the nondegenerate Trace form of a Lie algebra representation givesNondegeneracy therefore forces $\beta =-\alpha$. The standard sl2 subalgebra associated with a root argument gives one-dimensional opposite root spaces with vectors $e\in {\mathfrak{g}}_{\alpha }$, $f\in {\mathfrak{g}}_{-\alpha }$ and $h_{\alpha }\in \mathfrak{t}$ satisfyingTheir span is a copy of ${\mathfrak{sl}}_2$.

SetThen $\dim \mathfrak{h}=\dim \mathfrak{t}-1$, $\mathfrak{t}=\mathbb{C}{h}_{\alpha }\oplus \mathfrak{h}$, and every element of $\mathfrak{h}$ commutes with $e$, $f$, and $\mathfrak{t}$. Hence $\mathfrak{h}$ is abelian andas a direct sum of commuting Lie algebras. This is the two-root decomposition with a nondegenerate trace form.

Let $r=\dim \mathfrak{h}$ and choose a dual basis of weights ${\epsilon }_1,\dots ,{\epsilon }_r\in {\mathfrak{h}}^{\ast }$. TakeThe ${\mathfrak{sl}}_2$ factor acts in its defining representation on ${\mathbb{C}}^2$ and trivially on the other summands; $\mathfrak{h}$ acts trivially on ${\mathbb{C}}^2$ and by the displayed characters on the one-dimensional summands. The resulting trace form is the nondegenerate trace form on ${\mathfrak{sl}}_2$, ison $\mathfrak{h}$, and has zero cross terms. It is therefore nondegenerate on ${\mathfrak{sl}}_2\oplus \mathfrak{h}$.