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Symplectic Affine Action and Momentum with Cocycle

  • Augustin Batubenge
  • Wallace Haziyu
Chapter
Part of the STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health book series (STEAM)

Abstract

Let G be a Lie group, \(\mathfrak {g}\) its Lie algebra, and \(\mathfrak {g^{*}}\) the dual of \(\mathfrak {g}\). Let Φ be the symplectic action of G on a symplectic manifold (M, ω). If the momentum mapping \(\mu :M\rightarrow \mathfrak {g^{*}}\) is not Ad-equivariant, it is a fact that one can modify the coadjoint action of G on \(\mathfrak {g^{*}}\) in order to make the momentum mapping equivariant with respect to the new G-structure in \(\mathfrak {g^{*}}\), and the orbit of the coadjoint action is a symplectic manifold. With the help of a two cocycle \(\sum :\mathfrak {g}\times \mathfrak {g}\rightarrow \mathbf {R}\), \((\xi ,\eta )\mapsto \sum (\xi ,\eta )=d\hat {\sigma }_{\eta }(e)\cdot \xi \) associated with one cocycle \(\sigma :G\rightarrow \mathfrak {g^{*}};~~\sigma (g)=\mu (\phi _g(m))-Ad^*_g\mu (m)\), we show that a symplectic structure can be defined on the orbit of the affine action \(\Psi (g,\beta ):=Ad_{g}^{*}\beta +\sigma (g)\) of G on \(\mathfrak {g^{*}}\), the orbit of which is a symplectic manifold with the symplectic structure \(\omega _{\beta }(\xi _{\mathfrak {g^{*}}}(v),\eta _{\mathfrak {g^{*}}}(v))=-\beta ([\xi ,\eta ])+\sum (\eta ,\xi )\).

Furthermore, we introduce a deformed Poisson bracket on (M, ω) with which some classical results of conservative mechanics still hold true in a new setting.

Keywords

Symplectic action Momentum mapping Equivariance Poisson bracket 

Notes

Acknowledgements

Augustin Batubenge extends his thanks to Professor François Lalonde for his financial support during a stay at the University of Montreal in Summer 2016, which allowed the writing-up of this paper; along with Professor Norbert Hounkonnou for suggesting the use of a generalized Poisson bracket.

Wallace Haziyu is thankful to Doctors I.D. Tembo, M. Lombe, and A. Ngwengwe for the encouragements they rendered towards this project.

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Augustin Batubenge
    • 1
    • 2
  • Wallace Haziyu
    • 3
  1. 1.Départment de Mathématiques et StatistiqueUniversité de MontréalMontréalCanada
  2. 2.University of ZambiaLusakaZambia
  3. 3.Department of Mathematics and StatisticsUniversity of ZambiaLusakaZambia

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