Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 102 / 2

A finite root system in a real Euclidean vector space $E$ is a finite spanning set $\Phi \subset E\setminus \{0\}$ such that, for every $\alpha \in \Phi$, the root reflectionpreserves $\Phi$, and the Cartan integer $2(\beta ,\alpha )/(\alpha ,\alpha )$ is an integer for all $\alpha ,\beta \in \Phi$. It is reduced when the only scalar multiples of $\alpha$ in $\Phi$ are $\alpha$ and $-\alpha$. Its Weyl group is the subgroup of the orthogonal group generated by the reflections $s_{\alpha }$. A base of a root system $\Delta$ is a basis of $E$ such that every root is an integer combination of elements of $\Delta$ whose nonzero coefficients all have the same sign.

The coroot of $\alpha$ isLet $C$ be the Weyl chamber determined by $\Delta$:The roots $\alpha$ and ${\alpha }^{\vee }$ are positive scalar multiples, so their reflecting hyperplanes and their positive half-spaces are identical. The same chamber $C$ therefore defines positivity in the coroot system ${\Phi }^{\vee }$. Its walls correspond exactly to the rays ${\mathbb{R}}_{>0}{\alpha }^{\vee }$ for $\alpha \in \Delta$. Hence its simple roots arewhich is therefore a base of ${\Phi }^{\vee }$.

For an arbitrary finite-dimensional complex Lie algebra $L$, a Cartan subalgebra is a nilpotent Lie subalgebra $H$ equal to its own normalizer. When $L$ is semisimple, this is equivalently a maximal abelian subalgebra consisting of elements that act semisimply in the Adjoint representation.

Choose such an $H$. Simultaneous diagonalization gives the root-space decompositionThe nonzero weights $\alpha \in H^{\ast }$ are the roots. The restriction of the Killing form $B$ to $H$ is a nondegenerate bilinear form, so each $\alpha$ corresponds to a unique $t_{\alpha }\in H$ with $\alpha (h)=B(t_{\alpha },h)$. On the real span of these $t_{\alpha }$, the restriction of $B$ supplies a positive-definite inner product after choosing the standard real form. The sl2 subalgebra associated with a root givesand shows that these reflections preserve the finite set $\Phi$. Thus the roots form a finite reduced crystallographic root system, whose Weyl group is generated by these reflections.