拟三角、可分解Leibniz双代数与相对Rota-Baxter算子
Quasi-triangular, factorizable Leibniz bialgebras and relative Rota-Baxter operators
AI总结：
本文通过引入拟三角Leibniz双代数，利用带权相对Rota-Baxter算子刻画经典Leibniz Yang-Baxter方程的解，并建立斜对称二次Rota-Baxter Leibniz代数与三角及可分解Leibniz双代数之间的对应关系。
摘要：
我们引入拟三角Leibniz双代数的概念，它可以由经典Leibniz Yang-Baxter方程（CLYBE）的斜对称部分不变的解构造。除三角Leibniz双代数外，拟三角Leibniz双代数还包含可分解Leibniz双代数作为另一子类，后者导致底层Leibniz代数的分解。带权的相对Rota-Baxter算子被用来刻画斜对称部分不变的CLYBE解。在斜对称二次Leibniz代数上，这类算子对应于Rota-Baxter型算子。因此，我们引入斜对称二次Rota-Baxter Leibniz代数的概念，使得它们在权为$0$时产生三角Leibniz双代数，而在权非零时与可分解Leibniz双代数一一对应。
英文摘要：
We introduce the notion of quasi-triangular Leibniz bialgebras, which can be constructed from solutions of the classical Leibniz Yang-Baxter equation (CLYBE) whose skew-symmetric parts are invariant. In addition to triangular Leibniz bialgebras, quasi-triangular Leibniz bialgebras contain factorizable Leibniz bialgebras as another subclass, which lead to a factorization of the underlying Leibniz algebras. Relative Rota-Baxter operators with weights on Leibniz algebras are used to characterize solutions of the CLYBE whose skew-symmetric parts are invariant. On skew-symmetric quadratic Leibniz algebras, such operators correspond to Rota-Baxter type operators. Consequently, we introduce the notion of skew-symmetric quadratic Rota-Baxter Leibniz algebras, such that they give rise to triangular Leibniz bialgebras in the case of weight $0$, while they are in one-to-one correspondence with factorizable Leibniz bialgebras in the case of nonzero weights.
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