AI 中文总结
本文解决由李双代数和置换双代数构造莱布尼茨双代数的问题，证明二次李代数与置换双代数的张量积具有对应性质的莱布尼茨双代数结构，还构造了无限维莱布尼茨双代数并研究其拟三角、三角性质。
AI 中文摘要
本文研究由李双代数和置换双代数构造莱布尼茨双代数的问题。我们证明李代数与置换代数的张量积上存在莱布尼茨代数结构，并将该结论提升至双代数层面；进一步证明二次李代数与置换双代数的张量积具有莱布尼茨双代数结构，且若原置换双代数是上边缘（余边界）、拟三角、三角或可因子化的，则该莱布尼茨双代数也对应具有相同性质。此外，利用有限维李双代数与二次ℤ-分次置换代数的张量积构造了无限维莱布尼茨双代数，并对拟三角和三角的无限维莱布尼茨双代数进行了研究。
英文摘要
The construction problem of Leibniz bialgebras from Lie bialgebras and perm bialgebras is considered in this paper. We show that there is a Leibniz algebra structure on the tensor product of a Lie algebra and a perm algebra, and elevate this conclusion to the level of bialgebra. We prove that the tensor product of a quadratic Lie algebra and a perm bialgebra has a Leibniz bialgebra structure, and this Leibniz bialgebra structure is coboundary (resp. quasi-triangular, triangular, factorizable) if the original perm bialgebra is coboundary (resp. quasi-triangular, triangular, factorizable). Moreover, we constructed an infinite-dimensional Leibniz bialgebra using the tensor product of a finite-dimensional Lie bialgebra and a quadratic $\bz$-graded perm algebra. Quasi-triangular and triangular infinite-dimensional Leibniz bialgebras are considered.
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