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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关于辛李群的切线李群

通过阿斯卡里和萨利米蒙哈达（雷德保监会垫巴勒莫II SER 67：185-195，2018）的最近的工作动机上切线李群的黎曼几何，我们证明了切线李群\（{TG} \）一个的辛李群\（（G，\ omega）\）承认辛李群的结构。在\（{TG} \）上，我们使用G上左不变矢量场的完整和垂直提升，构造了一个左不变辛辛式\（{\ widetilde {\ omega}} \），它是从\（\ omega \）导出的。可以将上述构造视为在Asgari和Salimi Moghaddam中构造的切线Lie组上左不变黎曼度量的辛模拟（Rend Circ Mat Palermo II Ser 67：185–195，2018）。我们构造的一个直接结果是，通过获取非阿贝尔辛Lie群的迭代切线束，可以方便地生成任意高维的非阿贝尔辛Lie群。