Do It Yourself: the Structure Constants for Lie Algebras of Types El

Abstract

Two algorithms for computing the structure table of Lie algebras of type E l with respect to a Chevalley base are compared: the usual inductive algorithm and an algorithm based on the use of the Frenkel-Kac cocycle. It turns out that the Frenkel–Kac algorithm is several dozen times faster, but under the “natural” choice of a bilinear form and a sign function it has no success in a positive Chevalley base. We show how one can modify the sign function to obtain a proper choice of the structure constants. Cohen, Griess, and Lisser obtained a similar result by varying the bilinear form. We recall the hyperbolic realization of the root systems of type E l , which dramatically simplifies calculations as compared with the usual Euclidean realization. We give Mathematica definitions, which realize root systems and implement the inductive and Frenkel–Kac algorithms. Using these definitions, one can compute the whole structure table for E8 in a quarter of an hour with a home computer. At the end of the paper, we reproduce tables of roots ordered in accordance with HeightLex and the resulting tables of structure constants. Bibliography: 43 titles.