2016/06/30 by Lisa Carbone, Alex J. Feingold, Walter Freyn
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Algebra over a field #Algebraic structures and combinatorial models #Computer science #Embedding #Group (periodic table) #Mathematics #Nonlinear Waves and Solitons #Physics #Pure mathematics #Quantum mechanics #math.GR #math.RT #msc:20E42 #msc:20F05 #msc:20G44 #msc:51E24
paper · pdf · doi:10.3842/sigma.2020.045
published as SIGMA 16 (2020), 045, 47 pages · 47 pages, Latex with eight pdf figures, final published version
arxiv created 2020/05/29 · openalex publication_date 2020/05/29 · arxiv updated 2020/06/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
A be a symmetrizable hyperbolic generalized Cartan matrix with Kac-Moody algebra g = g(A) and (adjoint) Kac-Moody group G = G(A) = exp(ad(te i )), exp(ad(tf i )) | t C where e i and f i are the simple root vectors. Let B + , B -, N be the twin BN -pair naturally associated to G and let B + , B -be the corresponding twin building with Weyl group W and natural G-action, which respects the usual W -valued distance and codistance functions. This work connects the twin building B + , B -of G and the Kac-Moody algebra g = g(A) in a new geometrical way. The Cartan-Chevalley involution, , of g has fixed point real subalgebra, k, the 'compact' (unitary) real form of g, and k contains the compact Cartan t = k h. We show that a real bilinear form (, ) is Lorentzian with signatures (1, ) on k, and (1, n -1) on t. We define k k | (k, k) 0 to be the lightcone of k, and similarly for t. Let K be the compact (unitary) real form of G, that is, the fixed point subgroup of the lifting of to G. We construct a K-equivariant embedding of the twin building of G into the lightcone of the compact real form k of g. Our embedding gives a geometric model of part of the twin building, where each half consists of infinitely many copies of a W -tessellated hyperbolic space glued together along hyperplanes of the faces. Locally, at each such face, we find an SU(2)-orbit of chambers stabilized by U(1) which is thus parametrized by a Riemann sphere SU(2)/U(1) = S 2 . For n = 2 the twin building is a twin tree. In this case, we construct our embedding explicitly and we describe the action of the real root groups on the fundamental twin apartment. We also construct a spherical twin building at infinity, and construct an embedding of it into the set of rays on the boundary of the lightcone.