email@example.com, Anja.Steinbach@math.uni-giessen.de, firstname.lastname@example.org, email@example.com
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
Full text: dvi.gz 57 k, dvi 149 k, ps.gz 149 k, pdf.gz 192 k.