“Mathematics requires a small dose, not of genius, but of an imaginative freedom which, in a larger dose, would be insanity. And if mathematicians tend to burn out early in their careers, it is probably because life has forced them to acquire too much common sense, thereby rendering them too sane to work. But by then they are sane enough to teach, so a use can still be found for them.”
Angus K. Rodgers
For this week, I’ll write about the theory of root systems, especially things related to Weyl groups, weight spaces and etc, for the purpose of my URA. This is because one of the objectives of my URA is to generalise my previous blog post about a fraction of an \(S_n\)-orbit on a hyperplane to Weyl groups. First, a Lie algebra is simple if it is of degree greater than \(1\) (so we are excluding the one dimensional abelian Lie algebras) and has no proper ideals, and a semisimple Lie algebra is a direct sum of simple Lie algebras. Equivalently, a Lie algebra \(\mathfrak{g}\) is semisimple if \(\mathrm{rad}(\mathfrak{g})\), the sum of all solvable ideals (or the maximal solvable ideal), is trivial. Any semisimple Lie algebra \(\mathfrak{g}\) is the Lie algebra of an algebraic group \(G\). Let \(\mathrm{ad}:\mathfrak{g}\rightarrow \mathfrak{gl}(\mathfrak{g})\) by \((\mathrm{ad} X)(Z)=[X,Z]\) be the adjoint representation. Let \(B:\mathfrak{g}\times \mathfrak{g}\rightarrow \mathbf C\) be the symmetric bilinear form defined by \(B(X,Y)=\mathrm{tr}(\mathrm{ad}X\mathrm{ad} Y)\), called the Killing form which is invariant in the sense of \(B([X,Y],Z)=B(X,[Y,Z])\). Cartan’s criterion for semisimplicity says that a Lie algebra is semisimple iff the associated Killing form is nondegenerate. From now on, we let \(\mathfrak{g}\) be a semisimple Lie algebra over \(\mathbf C\). We start by discussing the root–space decomposition \(\mathfrak{g}=\mathfrak{h}\oplus \bigoplus_{\alpha\in\Phi}\mathfrak{g}_{\alpha}\) of semisimple Lie algebras.