Year: 2014
Type: article
Abstract: Let be a Dynkin diagram and let be the simple roots of the corresponding Kac–Moody root system. Let denote the Cartan subalgebra, let W denote the Weyl group and let Δ denote the set of all roots. The action of W on , and hence on Δ , is the discretization of the action of the Kac–Moody algebra. Understanding the orbit structure of W on Δ is crucial for many physical applications. We show that for , the simple roots α i and α j are in the same W –orbit if and only if vertices i and j in the Dynkin diagram corresponding to α i and α j are connected by a path consisting only of single edges. We introduce the notion of 'the Cayley graph of the Weyl group action on real roots' whose connected components are in one-to-one correspondence with the disjoint orbits of W . For a symmetric hyperbolic generalized Cartan matrix A of rank we prove that any two real roots of the same length lie in the same W –orbit. We show that if the generalized Cartan matrix A contains zeros, then there are simple roots that are stabilized by simple root reflections in W , that is, W does not act simply transitively on real roots. We give sufficient conditions in terms of the generalized Cartan matrix A (equivalently ) for W to stabilize a real root. Using symmetry properties of the imaginary light cone in the hyperbolic case, we deduce that the number of W –orbits on imaginary roots on a hyperboloid of fixed radius is bounded above by the number of root lattice points on the hyperboloid that intersect the closure of the fundamental region for W .
Authors: Lisa Carbone, Alexander Conway, Walter Freyn, Diego Penta
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
Funder: National Science Foundation
OpenAlex ID: W2094500618