反对称无穷小双代数的若干构造及其在Lie双代数中的应用
Some constructions of antisymmetric infinitesimal bialgebras and their applications to Lie bialgebras
- School of Mathematics and Statistics, Henan University（河南大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出利用Zinbiel双代数和双结合双代数构造反对称无穷小双代数及Lie双代数的方法，并证明其保持拟三角、三角和可分解性质。
AI中文摘要：
本文主要提供了利用Zinbiel双代数和双结合双代数构造反对称无穷小（ASI）双代数及Lie双代数的方法。我们首先证明，在一个双结合双代数与一个二次Zinbiel代数的张量积上存在ASI双代数结构，而在一个Zinbiel双代数与一个二次$\z$-分次双结合代数的张量积上存在无穷维ASI双代数结构。对于特殊的二次$\z$-分次Leibniz代数，其与Zinbiel双代数的张量积构成ASI双代数的性质刻画了该Zinbiel双代数。通过考察Zinbiel代数中（经典）Yang-Baxter方程的解与其诱导的结合代数之间的关系，我们证明了当原Zinbiel双代数是拟三角（resp. 三角，可分解）时，诱导的ASI双代数也是拟三角（resp. 三角，可分解）。这些结论使我们能够提供一种从Zinbiel双代数构造Lie双代数的方法，以及两种从双结合双代数构造Lie双代数的方法。我们还具体描述了这些构造所对应的Yang-Baxter方程解之间的联系以及$O$-算子之间的联系。
英文摘要：
In this paper, we mainly provide some methods for constructing antisymmetric infinitesimal (ASI) bialgebras and Lie bialgebras using Zinbiel bialgebras and diassociative bialgebras. We first show that there is an ASI bialgebra structure on the tensor product of a diassociative bialgebra and a quadratic Zinbiel algebra, while there is an infinite-dimensional ASI bialgebra structure on the tensor product of a Zinbiel bialgebra and a quadratic $\bz$-graded diassociative algebra. For a special quadratic $\bz$-graded Leibniz algebra, the property that its tensor product with a Zinbiel bialgebra forms an ASI bialgebra characterizes the Zinbiel bialgebra. By examining the relationship between solutions of the (classical) Yang-Baxter equation in a Zinbiel algebra and the induced associative algebra, we prove that the induced ASI bialgebra is quasi-triangular (resp. triangular, factorizable) whenever the original Zinbiel bialgebra is quasi-triangular (resp. triangular, factorizable). These conclusions enable us to provide a method for constructing Lie bialgebras from Zinbiel bialgebras and two approaches for constructing Lie bialgebras from diassociative bialgebras. We also provide specific descriptions of the connections between the solutions of the Yang-Baxter equations and the connections between the $O$-operators corresponding to these constructions.
