AI 中文总结
本文证明反对称无穷小双代数与二次 perm 代数的张量积具有双结合双代数结构，且其结构性质与原反对称无穷小双代数对应，还明确了几类双代数及辛代数间的密切关系。
AI 中文摘要
结合代数与 perm 代数的张量积上存在双结合代数结构。本文将该结论提升至双代数层面，证明反对称无穷小双代数与二次 perm 代数的张量积具有双结合双代数结构，且若原反对称无穷小双代数是上边缘（余边界）、拟三角、三角、可分解的，则该双结合双代数结构对应为上边缘（余边界）、拟三角、三角、可分解的。作为应用，本文给出李双代数、莱布尼茨双代数、双结合双代数与反对称无穷小双代数之间的密切关系，以及辛李代数、辛莱布尼茨代数、辛结合代数与辛双结合代数之间的密切关系。
英文摘要
There is a diassociative algebra structure on the tensor product of an associative algebra and a perm algebra. In this paper, we elevate this conclusion to the level of bialgebra.We prove that the tensor product of an antisymmetric infinitesimal bialgebra and a quadratic perm algebra has a diassociative bialgebra structure, and this diassociative bialgebra structure is coboundary (resp. quasi-triangular, triangular, factorizable) if the original antisymmetric infinitesimal bialgebra is coboundary (resp. quasi-triangular,triangular, factorizable). As an application, we provide the close relationship between Lie bialgebras, Leibniz bialgebras, diassociative bialgebras and antisymmetric infinitesimal bialgebras, and the close relationship between symplectic Lie algebras, symplectic Leibniz algebras, symplectic associative algebras and symplectic diassociative algebras.
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