On the tangent Lie group of a symplectic Lie group

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.

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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)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Baues, O., Cortés, V.: Symplectic Lie groups I–III. arXiv:1307.1629v2

  3. 3.

    Lichnerowicz, A., Medina, A.: On Lie groups with left-invariant symplectic or Kählerian structures. Lett. Math. Phys. 16(3), 225–235 (1988)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Yano, K., Ishihara, S.: Tangent and Cotangent Bundles. Marcel Decker, Inc., New York (1973)

    Google Scholar 

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Correspondence to David N. Pham.

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Pham, D.N. On the tangent Lie group of a symplectic Lie group. Ricerche mat 68, 699–704 (2019). https://doi.org/10.1007/s11587-019-00434-2

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Keywords

  • Symplectic Lie groups
  • Tangent Lie groups
  • Vertical lifts
  • Complete lifts

Mathematics Subject Classification

  • 22E15
  • 53D015