Journal of Generalized Lie Theory and Applications

Canonical endomorphism field on a Lie algebra


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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.

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J. Gen. Lie Theory Appl., Volume 4 (2010), Article ID G100302, 17 pages.

First available in Project Euclid: 11 October 2011

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Primary: 17B08: Coadjoint orbits; nilpotent varieties 53C15: General geometric structures on manifolds (almost complex, almost product structures, etc.) 53C80: Applications to physics 70G45: Differential-geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) [See also 53Cxx, 53Dxx, 58Axx] 70G60: Dynamical systems methods 70H03: Lagrange's equations 70H05: Hamilton's equations


KOCIK, Jerzy. Canonical endomorphism field on a Lie algebra. J. Gen. Lie Theory Appl. 4 (2010), Article ID G100302, 17 pages. doi:10.4303/jglta/G100302.

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