Geodesic
Symplectic Lie groups and doubled geometry
David N. Pham1 ; Fei Ye1 ; David N. Pham ; Fei Ye
1Queensborough C. College City University of New York, 56th Avenue Bayside, NY 11364, U.S.A
Acta mathematica Universitatis Comenianae, Tome 91 (2022) no. 2, pp. 161-190 
David N. Pham; Fei Ye; David N. Pham; Fei Ye. Symplectic Lie groups and doubled geometry. Acta mathematica Universitatis Comenianae, Tome 91 (2022) no. 2, pp. 161-190. http://geodesic.mathdoc.fr/item/AMUC_2022_91_2_a5/
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     author = {David N. Pham and Fei Ye and David N. Pham and Fei Ye},
     title = { Symplectic {Lie} groups and doubled geometry},
     journal = {Acta mathematica Universitatis Comenianae},
     pages = {161--190},
     year = {2022},
     volume = {91},
     number = {2},
     url = {http://geodesic.mathdoc.fr/item/AMUC_2022_91_2_a5/}
}
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Voir la notice de l'article provenant de la source Comenius University

A left invariant flat para-Kähler structure is constructed on the tangent Lie group of a symplectic Lie group (G, ω). Remarkably, it is shown that the left invariant para-Kähler form on TG coincides with a certain pullback of the standard symplectic form on T∗G. The immediate upshot of this is that T∗G can be equipped with a Lie group structure for which the standard symplectic form is left invariant. Lastly, the double field theory geometry of the double manifold TG is studied using the geometric framework of Vaisman [22, 23].