Geometry of Tangent Poisson-Lie Groups

被引：0

作者：

Al-Dayel, Ibrahim ^{[1
]}

Aloui, Foued ^{[1
]}

Deshmukh, Sharief ^{[2
]}

机构：

[1] Imam Mohammad Ibn Saud Islamic Univ IMSIU, Coll Sci, Dept Math & Stat, POB 65892, Riyadh 11566, Saudi Arabia

[2] King Saud Univ, Dept Math, Riyadh 11495, Saudi Arabia

来源：

MATHEMATICS | 2023年 / 11卷 / 01期

关键词：

Poisson geometry; Riemannian geometry; Lie group; Lie algebra; COMPATIBILITY; BUNDLE;

D O I：

10.3390/math11010240

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

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 TG of G. In this paper, we induce a left invariant contravariant pseudo-Riemannian metric on the tangent bundle TG, and we express in different cases the contravariant Levi-Civita connection and curvature of TG in terms of the contravariant Levi-Civita connection and the curvature of G. We prove that the space of differential forms omega*(G) on G is a differential graded Poisson algebra if, and only if, omega*(TG) 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 TG is also a pseudo-Riemannian Poisson-Lie group. Finally, some examples of pseudo-Riemannian tangent Poisson-Lie groups are given.

引用

页数：18