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Geometry of Tangent Poisson–Lie Groups

Author

Listed:
  • Ibrahim Al-Dayel

    (Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), P.O. Box 65892, Riyadh 11566, Saudi Arabia)

  • Foued Aloui

    (Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), P.O. Box 65892, Riyadh 11566, Saudi Arabia)

  • Sharief Deshmukh

    (Department of Mathematics, King Saud University, Riyadh 11495, Saudi Arabia)

Abstract

Let G be a Poisson–Lie group equipped with a left invariant contravariant pseudo-Riemannian metric. There are many ways to lift the Poisson structure on G to the tangent bundle T G of G . In this paper, we induce a left invariant contravariant pseudo-Riemannian metric on the tangent bundle T G , and we express in different cases the contravariant Levi-Civita connection and curvature of T G in terms of the contravariant Levi-Civita connection and the curvature of G . We prove that the space of differential forms Ω * ( G ) on G is a differential graded Poisson algebra if, and only if, Ω * ( T G ) is a differential graded Poisson algebra. Moreover, we show that G is a pseudo-Riemannian Poisson–Lie group if, and only if, the Sanchez de Alvarez tangent Poisson–Lie group T G is also a pseudo-Riemannian Poisson–Lie group. Finally, some examples of pseudo-Riemannian tangent Poisson–Lie groups are given.

Suggested Citation

  • Ibrahim Al-Dayel & Foued Aloui & Sharief Deshmukh, 2023. "Geometry of Tangent Poisson–Lie Groups," Mathematics, MDPI, vol. 11(1), pages 1-18, January.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:1:p:240-:d:1023393
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