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On the tangent Lie group of a symplectic Lie group

  • David N. PhamEmail author
Article
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Abstract

Motivated by the recent work of Asgari and Salimi Moghaddam (Rend Circ Mat Palermo II Ser 67:185–195, 2018) on the Riemannian geometry of tangent Lie groups, we prove that the tangent Lie group \({ TG}\) of a symplectic Lie group \((G,\omega )\) admits the structure of a symplectic Lie group. On \({ TG}\), we construct a left invariant symplectic form \({\widetilde{\omega }}\) which is induced from \(\omega \) using complete and vertical lifts of left invariant vector fields on G. The aforementioned construction can be viewed as the symplectic analogue of the left invariant Riemannian metrics on the tangent Lie groups that were constructed in Asgari and Salimi Moghaddam (Rend Circ Mat Palermo II Ser 67:185–195, 2018). One immediate upshot of our construction is that by taking iterated tangent bundles of a non-abelian symplectic Lie group, one obtains a convenient means of generating non-abelian symplectic Lie groups of arbitrarily high dimension.

Keywords

Symplectic Lie groups Tangent Lie groups Vertical lifts Complete lifts 

Mathematics Subject Classification

22E15 53D015 

Notes

References

  1. 1.
    Asgari, F., Salimi Moghaddam, H.R.: On the Riemannian geometry of tangent Lie groups. Rend. Circ. Mat. Palermo II Ser. 67, 185–195 (2018)MathSciNetzbMATHGoogle Scholar
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    Baues, O., Cortés, V.: Symplectic Lie groups I–III. arXiv:1307.1629v2
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    Lichnerowicz, A., Medina, A.: On Lie groups with left-invariant symplectic or Kählerian structures. Lett. Math. Phys. 16(3), 225–235 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
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    Yano, K., Ishihara, S.: Tangent and Cotangent Bundles. Marcel Decker, Inc., New York (1973)zbMATHGoogle Scholar

Copyright information

© Università degli Studi di Napoli "Federico II" 2019

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceQCC CUNYBaysideUSA

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