Do it yourself structure constants for Lie algebras of types~$\mathrm E_l$

N. A. Vavilov

Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 8, Tome 281 (2001), pp. 60-104

Résumé

We compare two algorithms to compute structure table of the Lie algebras of type $\mathrm E_l$ with respect to a Chevalley base: the usual inductive algorithm and an algorithm based on the use of Frenkel–Kac cocycle. It turns out that Frenkel–Kac algorithm is several dozen times faster but with the ‘natural’ choice of the bilinear form and the sign function it does not give the result in a positive Chevalley base. We show how to modify the sign function to get the right choice of structure constants. Cohen, Griess and Lisser obtained similar result by varying the bilinear form. We recall the hyperbolic realisation of root systems of type $\mathrm E_l$ which dramatically simplify calculations as compared with the usual Euclidean realisation. We give Matematica definitions which realise root systems and implement both the inductive and Frenkel–Kac algorithms. Using these definitions one can compute the whole structure table for $\mathrm E_8$ within a quarter of an hour at a home computer. At the end of the paper we reproduce tables of roots according to HeightLex and the resulting tables of structure constants.