# 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

## References

1. 1.
A. Kirillov, Introduction to Lie Groups and Lie Algebras. Department of Mathematics (Suny at Stony Brook, New York, 2008)Google Scholar
2. 2.
R. Abraham, J.E. Marsden, Foundations of Mechanics, 2nd edn. (The Benjamin/Cummings Publishing Company, Reading, MA, 1978)
3. 3.
R. Berndt, An Introduction to Symplectic Geometry, vol. 26 (American Mathematical Society, Rhodes Island, 2001)
4. 4.
P. Iglesias-Zemmour, Diffeology. Mathematical Surveys and Monographs, vol. 185 (American Mathematical Society, Providence, RI, 2010)Google Scholar
5. 5.
F.W. Warner, Foundations of Differentiable Manifolds and Lie Groups (Springer, New York, 1983)
6. 6.
J. Śniatycki, Differential Geometry of Singular Spaces and Reduction of Symmetry. New Mathematical Monographs, vol. 23 (Cambridge University Press, Cambridge, 2013)Google Scholar

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## 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