Linear Poisson Structures and Lie Algebras

  • Camille Laurent-Gengoux
  • Anne Pichereau
  • Pol Vanhaecke
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 347)


Together with symplectic manifolds, considered in the previous chapter, Lie algebras provide the first examples of Poisson manifolds. The dual \(\mathfrak{g}^{*}\) of a finite-dimensional Lie algebra \(\mathfrak{g}\) admits a natural Poisson structure, called its Lie–Poisson structure. It is a linear Poisson structure and every linear Poisson structure (on a finite-dimensional vector space) is a Lie–Poisson structure. We show that the leaves of the symplectic foliation are the coadjoint orbits of the adjoint group of \(\mathfrak{g}\) and we shortly discuss the linearization of Poisson structures (in the neighborhood of a point where the rank is zero). Using a non-degenerate Ad-invariant symmetric bilinear form, we get the Lie-Poisson structure on \(\mathfrak{g}\), which has several virtues, amongst which the fact that the Hamiltonian vector fields on \(\mathfrak{g}\) take a natural form, a so-called Lax form. Affine Poisson structures and their Lie theoretical interpretation are discussed at the end of the chapter.


Modular Form Poisson Bracket Poisson Structure Coadjoint Orbit Poisson Manifold 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Camille Laurent-Gengoux
    • 1
  • Anne Pichereau
    • 2
  • Pol Vanhaecke
    • 3
  1. 1.CNRS UMR 7122, Laboratoire de MathématiquesUniversité de LorraineMetzFrance
  2. 2.CNRS UMR 5208, Institut Camille JordanUniversité Jean MonnetSaint-EtienneFrance
  3. 3.CNRS UMR 7348, Lab. Mathématiques et ApplicationsUniversité de PoitiersFuturoscope ChasseneuilFrance

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