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Deformation of the Poisson structure related to algebroid bracket of differential forms and application to real low dimentional Lie algebras. (English) Zbl 1465.17021

Mladenov, Ivaïlo M. (ed.) et al., Proceedings of the 20th international conference on geometry, integrability and quantization, Sts. Constantine and Elena (near Varna), Bulgaria, June 2–7, 2018. Sofia: Avangard Prima; Sofia: Bulgarian Academy of Sciences, Institute of Biophysics and Biomedical Engineering. Geom. Integrability Quantization 20, 122-130 (2019).
This paper studies classification of some real low dimensional Lie algebras from the perspective of Poisson geometry and Lie algebroid formalisms. The Casimir functions are calculated. By vertically lifting of the Poisson vector fields from Poisson manifold to its tangent bundle, new Poisson tensors and classes of bi-Hamiltonian manifolds are obtained (this method is also known as deformations of Poisson structures using the bi-Hamiltonian structure). Application to a few examples of low dimensional Lie algebras yields new invariants. For example, by taking the three dimensional nilpotent Lie algebra \(A_{3,1}\), one could generate nilpotent Lie algebras of dimension six.
For the entire collection see [Zbl 1408.53003].

MSC:

17B63 Poisson algebras
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
53D17 Poisson manifolds; Poisson groupoids and algebroids
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Full Text: Euclid