Super Weyl Groups for Basic Classical Lie Superalgebras via Odd Root Parity — E8 Intelligence Research

FINDING: Super Weyl groups for basic classical Lie superalgebras are defined via odd root parity relations and directed Coxeter graphs, extending classical Weyl group theory to superalgebras. | MATH: The super Weyl group is a quotient of the classical Weyl group by relations encoding odd root parity (e.g., \(s_\alpha^2 = 1\) for even roots, but \(s_\alpha^2 \neq 1\) for odd roots in some cases). Directed edges in the Coxeter graph indicate non-standard braid relations (e.g., \(m_{ij} = 2,3,4,6\) with parity-dependent signs). Key constants: root system Cartan matrices with entries \(a_{ij} = 2(\alpha_i,\alpha_j)/(\alpha_i,\alpha_i)\); parity vector \(\epsilon_i = 0\) (even) or \(1\) (odd). | CONNECTION: Root systems of basic classical Lie superalgebras (e.g., \(A(m,n), B(m,n), C(n), D(m,n), D(2,1;\alpha), F(4), G(3)\)) exhibit crystallographic symmetries akin to classical root systems but with odd roots introducing "twisted" reflections. The Coxeter graph directed edges mirror the asymm Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com