Oscillator Lie algebras are the only non commutative solvable Lie algebras which carry a bi-invariant Lorentzian metric. In this paper, we determine all the Poisson structures, and in particular, all symmetric Leibniz algebra structures whose underlying Lie algebra is an oscillator Lie algebra. We give also all the symmetric Leibniz bialgebra structures whose underlying Lie bialgebra structure is a Lie bialgebra structure on an oscillator Lie algebra. We derive some geometric consequences on oscillator Lie groups.

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振荡器李代数上的泊松代数和对称莱布尼茨双代数结构

振荡器李代数是唯一带有双不变洛伦兹度量的非交换可解李代数。在本文中，我们确定所有 Poisson 结构，特别是所有对称 Leibniz 代数结构，其基础李代数是振荡器李代数。我们还给出了所有对称的莱布尼茨双代数结构，其基础李双代数结构是振荡器李代数上的李双代数结构。我们推导出一些关于振荡器李群的几何结果。