Zeitschrift für verallgemeinerte Lügentheorie und -anwendungen

Canonical endomorphism eld on a Lie algebra

Abstract

Jerzy KOCIK

We show that every Lie algebra is equipped with a natural (1; 1)-variant tensor eld, the \canonical endomorphism eld", determined by the Lie structure, and satisfying a certain Nijenhuis bracket condition. This observation may be considered as complementary to the Kirillov-Kostant-Souriau theorem on symplectic geometry of coadjoint orbits. We show its relevance for classical mechanics, in particular for Lax equations. We show that the space of Lax vector elds is closed under Lie bracket and we introduce a new bracket for vector elds on a Lie algebra. This bracket denes a new Lie structure on the space of vector elds.