Canonical endomorphism eld on a Lie algebraJerzy KOCIK
Department of Mathematics, Southern Illinois University, Carbondale, IL 62901, USA E-mail: [email protected]
Received Date: March 15, 2010; Revised Date: April 28, 2010
Notation. We will distinguish between purely algebraic and Differential products by using two types of brackets:
: Lie algebra product,
[ , ]: Lie commutator of vectorfields, Schouten bracket, Nijenhuis bracket.
The summation convention over repeated indices is adopted throughout the paper.
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 denes a new Lie structure on the space of vector elds.