拟三角Jordan D-双代数与扩展的相对Rota-Baxter算子
Quasi-triangular Jordan D-bialgebras and extended relative Rota-Baxter operators
AI总结：
本文通过引入扩展的相对Rota-Baxter算子建立Jordan Yang-Baxter方程的算子研究框架，构造拟三角与可因子化Jordan D-双代数，揭示二次Rota-Baxter Jordan代数与两类双代数的对应关系。
AI中文摘要：
本文引入了拟三角Jordan D-双代数，这类双代数由对称部分具有不变性的Jordan Yang-Baxter方程（JYBE）的解构造而成。我们首先通过在Jordan代数上引入扩展的相对Rota-Baxter算子的概念，建立了研究JYBE的系统化算子方法。研究表明，这类算子及其扩展可以诱导出新的Jordan代数结构。此外，线性映射的对称化子-反对称化子分解将扩展的相对Rota-Baxter算子的研究分别简化为Jordan代数同态对和相对Rota-Baxter算子的研究。该算子框架随后给出了对称部分具有不变性的JYBE解的一种刻画。我们分别在二次Jordan代数和半直积Jordan代数的背景下进一步研究了这些算子形式，得到了JYBE解的显式构造。接着，本文引入了可因子化Jordan D-双代数作为拟三角Jordan D-双代数的一类特殊情形，它能自然地将底层的Jordan代数因子化。我们证明，任意Jordan D-双代数的Drinfeld经典双代数都具有可因子化Jordan D-双代数结构。最后，从算子视角出发，本文提出了二次Rota-Baxter Jordan代数的概念：零权的二次Rota-Baxter Jordan代数可生成三角Jordan D-双代数，而非零权的二次Rota-Baxter Jordan代数则与可因子化Jordan D-双代数一一对应。
英文摘要：
This paper introduces quasi-triangular Jordan D-bialgebras, which are constructed from solutions of the Jordan Yang-Baxter equation (JYBE) with invariant symmetric parts. We first develop a systematic operator approach to the JYBE by introducing the notion of extended relative Rota-Baxter operators on Jordan algebras. Such operators with their extensions are shown to induce new Jordan algebra structures. Moreover, the symmetrizer-antisymmetrizer decomposition of linear maps reduces the study of extended relative Rota-Baxter operators to both pairs of homomorphisms of Jordan algebras and relative Rota-Baxter operators, respectively. This operator framework subsequently yields a characterization of solutions of the JYBE whose symmetric parts are invariant. These operator forms are further investigated in the context of quadratic Jordan algebras and semi-direct product Jordan algebras, respectively, leading to explicit constructions of solutions of the JYBE. A factorizable Jordan D-bialgebra is then introduced as a special class of quasi-triangular ones, which naturally factorizes the underlying Jordan algebra. We prove that the Drinfeld classical double of any Jordan D-bialgebra admits a factorizable Jordan D-bialgebra structure. Finally, the operator perspective gives rise to the notion of quadratic Rota-Baxter Jordan algebras: those of zero weight yield triangular Jordan D-bialgebras, while those of nonzero weight are shown to be in one-to-one correspondence with factorizable Jordan D-bialgebras.
