Journal of Generalized Lie Theory and Applications

Canonical endomorphism field on a Lie algebra

Jerzy KOCIK

Full-text: Open access

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.

Article information

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

Dates
First available in Project Euclid: 11 October 2011

Permanent link to this document
https://projecteuclid.org/euclid.jglta/1318365489

Digital Object Identifier
doi:10.4303/jglta/G100302

Mathematical Reviews number (MathSciNet)
MR2795573

Zentralblatt MATH identifier
1298.70022

Subjects
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

Citation

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. https://projecteuclid.org/euclid.jglta/1318365489


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