Canonical endomorphism field on a Lie algebra

TL;DR

This paper introduces a natural tensor field on Lie algebras called the canonical endomorphism field, explores its properties, and demonstrates its relevance to classical mechanics and Lax equations.

Contribution

It defines the canonical endomorphism field on any Lie algebra and investigates its properties, including a new Lie bracket on vector fields related to Lax equations.

Findings

The canonical endomorphism field satisfies a Nijenhuis bracket condition.

The space of Lax vector fields is closed under a new Lie bracket.

A new Lie structure on vector fields on a Lie algebra is introduced.

Abstract

We show that every Lie algebra is equipped with a natural $(1,1)$-variant tensor field, the "canonical endomorphism field", naturally 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 fields is closed under Lie bracket and we introduce a new bracket for vector fields on a Lie algebra. This bracket defines a new Lie structure on the space of vector fields.