Weyl Group Fixed Points and One-Dimensional Root Spaces in Compact Lie Algebras — E8 Intelligence Research

FINDING: The Weyl group fixed-point subspace within a Cartan subalgebra defines the fundamental domain of root-system symmetry, and root spaces are 1-dimensional for compact semisimple groups — a structural rigidity that underpins all simple Lie algebra classifications. MATH: - Cartan subalgebra \\(\\mathfrak{h}\\) of semisimple \\(\\mathfrak{g}\\); root decomposition: \\(\\mathfrak{g} = \\mathfrak{h} \\oplus \\bigoplus_{\\alpha \\in \\Delta} \\mathfrak{g}_\\alpha\\), with \\(\\dim \\mathfrak{g}_\\alpha = 1\\) (for compact semisimple). - Weyl group \\(W = N_G(\\mathfrak{h})/Z_G(\\mathfrak{h})\\) acts on \\(\\mathfrak{h}^*\\); fixed-point subspace: \\(\\mathfrak{h}^W = \\{ h \\in \\mathfrak{h} : w \\cdot h = h \\ \\forall w \\in W \\}\\). - 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 \\Delta^+} \\alpha\\). - Frobenius Lie algebra principal element: \\(w = \\sum_i x_i y_i\\) (from Ge Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com