Weyl Group Action and Root Systems in Semisimple Lie Algebras — E8 Intelligence Research

FINDING: The Weyl group acts on the Cartan subalgebra's fixed-point subspace, and root systems encode the full symmetry of semisimple Lie algebras; the Weyl character formula and 1-dimensional root spaces are central structural results. | MATH: Cartan subalgebra \(\mathfrak{h} \subset \mathfrak{g}\); root system \(\Phi \subset \mathfrak{h}^*\); Weyl group \(W = N_G(\mathfrak{h})/Z_G(\mathfrak{h})\) acts on \(\mathfrak{h}\) with fixed-point subspace \(\mathfrak{h}^W\) (dimension = rank of \(\mathfrak{g}\)); root spaces \(\mathfrak{g}_\alpha\) are 1-dimensional for each \(\alpha \in \Phi\); Weyl character formula: \(\chi_\lambda = \frac{\sum_{w \in W} (-1)^{\ell(w)} e^{w(\lambda+\rho)}}{\sum_{w \in W} (-1)^{\ell(w)} e^{w(\rho)}}\), with \(\rho = \frac{1}{2}\sum_{\alpha \in \Phi^+} \alpha\). | CONNECTION: Root systems are crystallographic — all angles between roots are multiples of \(30^\circ\) or \(45^\circ\), yielding ratios \(\cos\theta \in \{0, \pm\tfrac12, \pm\tfrac{\sqrt2}{2}, \pm\t Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com