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Do it yourself: the structure constants for Lie algebras of types \(E_l\). (English. Russian original) Zbl 1062.17004

J. Math. Sci., New York 120, No. 4, 1513-1548 (2004); translation from Zap. Nauchn. Semin. POMI 281, 60-104 (2001).
From the text: This is the first of the proposed series of five publications devoted to a computer study of minimal modules for exceptional Chevalley groups. The goal of this work is to list explicitly the root elements \(x_\alpha(\xi)\), \(w_\alpha (\varepsilon)\), and \(h_\alpha (\varepsilon)\) and defining equations for the exceptional groups on these modules.
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. A. M. Cohen, R. L. Griess and B. Lisser [Commun. Algebra 21, No. 6, 1889–1907 (1993; Zbl 0805.20015)] 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 [I. B. Frenkel and V. Kac, Invent. Math. 62, No. 1, 23–66 (1980; Zbl 0493.17010)]. Using these definitions, one can compute the whole structure table for \(E_8\) 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. The tables coincide in essence with those in P. B. Gilkey and G. M. Seitz, Geom. Dedicata 25, No. 1–3, 407–416 (1988; Zbl 0661.17005).

MSC:

17B20 Simple, semisimple, reductive (super)algebras
20G20 Linear algebraic groups over the reals, the complexes, the quaternions
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