Lie algebras generated by extremal elements

amc@win.tue.nl, Anja.Steinbach@math.uni-giessen.de, rosane@win.tue.nl, dbw@caltech.edu

Submission: 1999, Mar. 29

We study Lie algebras L over a field of characteristic distinct from 2, which are generated by extremal elements (i.e., by elements x for which [x, [x, L]] is contained in kx). Lie algebras of Chevalley type, generated by the long root elements, are an example.

We construct an associative symmetric bilinear form on L and investigate its relationship with the Killing form. We prove that any Lie algebra generated by a finite number of extremal elements is finite dimensional.

The minimal number of extremal generators for the Lie algebras of type A_n (n >= 1), B_n (n >= 3), C_n (n >= 2), D_n (n >= 4), E_n (n = 6,7,8), F_4 and G_2 are shown to be n+1, n+1, 2n, n, 5, 5, and 4 in the respective cases. These results are related to group theoretic ones for the corresponding Chevalley groups.

1991 Mathematics Subject Classification: 17B45, 17B01, 20G15

Keywords and Phrases: Lie algebras, inner ideal, sandwich element, simple Lie algebra of Chevalley type

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