Geometry of Tangent Poisson–Lie Groups

Ibrahim Al-Dayel (), Foued Aloui and Sharief Deshmukh
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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

Mathematics, 2023, vol. 11, issue 1, 1-18

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.

Keywords: Poisson geometry; Riemannian geometry; Lie group; Lie algebra (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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