Algebra Seminar

We shall discuss Lie algebras L generated by extremal elements, that is elements x such that [x,[x,L]] \subseteq <x>. Any Lie algebra generated by a finite number of extremal elements is finite dimensional. The minimal number of extremal generators for the complex simple Lie algebras of type A_n (n>0), B_n (n>2), C_n (n>1), D_n (n>3), E_n (n=6,7,8), F₄ and G₂ are n+1, n+1, 2n, n, 5, 5, and 4 in the respective cases. There is an elegant structure theory for the Lie algebras under study, and there are connections with Tits geometry, notably Timmesfeld's classification of groups generated by abstract root groups.