Lie Algebras Generated by Extremal Elements

A Lie algebra L is a vector space over the field F accompanied by a bilinear map [·, ·]: L × L → L which is skew-symmetric (i.e. [x, x] = 0 for all x ∈ L) and satisfies the Jacobi identity: [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0 for all x, y, z ∈ L. Lie algebras have their applications for example in physics (see for example [SW86] or [BK90]), and in the study of differential equations (see for example the PhD thesis by Jan Draisma [Dra02]). An element x ∈ L is called an extremal element if [x, [x, L]] ⊆ Fx. In this Master’s thesis we study Lie algebras generated by finitely many extremal elements, building on results by Cohen, Steinbach, Ushirobira, and Wales [CSUW01]. In that paper various important properties of Lie algebras generated by extremal elements are proved, for example the fact that a Lie algebra generated by finitely many extremal elements is always finite dimensional. It is also proved that all simple Lie algebras can be generated by extremal elements. Moreover, the two and three extremal generator cases are extensively studied. i