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Canonical endomorphism field on a Lie algebra. (English) Zbl 1298.70022
Author’s abstract: We show that every Lie algebra is equipped with a natural $$(1,1)$$-variant tensor field, the “canonical endomorphism field”, 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.

Reviewer’s remark: Beyond the results obtained by straightforward computation there remain some questions: Integral manifolds of a distribution not having constant rank? “Adjoint orbits”? A $$(1,1)$$-tensor “acting” on these adjoint orbits? Supplementary explanations are needed!

MSC:
 70G45 Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics 17B66 Lie algebras of vector fields and related (super) algebras 53D17 Poisson manifolds; Poisson groupoids and algebroids
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