TRIALITY CONSTRUCTION OF EXCEPTIONAL LIE ALGEBR A S

Niveau: Supérieur, Licence, Bac+3
TRIALITY, CONSTRUCTION OF EXCEPTIONAL LIE ALGEBR A S Frédéric BUTIN 20th February 2006 In the classification of the root systems, the Dynkin diagram (G2) (Diagram 1) appears. According to Serre's theorem, there exists an only complex semisimple Lie algebra that admits this root system. We will give three constructions of this Lie algebra : • A very concrete version by calculating its multiplication table. • A construction from the representations of ?l3. • A construction from the octonion algebra. 1 First construction of g2 1.1 Using the Dynkin diagram The classification of the root systems shows the sys- tem given by Diagram 2, for which the Dynkin dia- gram is Diagram 1. Diagram 1 Diagram 2 Theorem 1 (Serre) Let ∆ be a root system, let ? be a basis of ∆, and (n(?, ?))?,??? its Cartan matrix. Then the Lie algebra presented by generators E?, F?, H?, ? ? ∆ and Weyl-Serre relations is a finite-dimensional semisimple Lie algebra, and its root system is ∆. According to Serre's theorem, there exists so a complex Lie algebra which has this root system. But we want to verify it by hand. Suppose that such a Lie algebra g2 exists. Since it has 12 roots, and since dimC h = dimR h?R = 2, this algebra is necessary of dimension 14=12+2.

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