AI 中文总结
本文研究莱布尼茨代数与约旦对双系的关联，给出约旦对双系的Tits-Kantor-Koecher构造，引入其同伦与莱布尼茨代数的阿贝尔内理想，扩展了约旦dialgebras的相关构造。
AI 中文摘要
本文提出了约旦对的一种推广，称为约旦对双系，它自然地作为具有有限ℤ-分次的莱布尼茨代数的 wings 出现；反之，我们为约旦对双系给出了Tits-Kantor-Koecher构造，得到一个具有短ℤ-分次的莱布尼茨代数。该构造扩展了Gubarev和Kolesnikov针对约旦dialgebras（arXiv:0907.1740）给出的构造。我们在元素处引入约旦对双系的同伦，并证明由此再次得到一个约旦dialgebra。此外，我们为莱布尼茨代数引入阿贝尔内理想及其核，证明莱布尼茨代数关于阿贝尔内理想的子商是约旦对双系，该子商概念扩展了Felipe和Velásquez针对Q-约旦元素的约旦dialgebras构造。
英文摘要
In this paper we present a generalization of Jordan pairs, called Jordan pair disystem, which appear naturally as the wings of a Leibniz algebra with a finite $\mathbb{Z}$-grading; conversely, we give the Tits-Kantor-Koecher construction for Jordan pair disystems, obtaining a Leibniz algebra with a short $\mathbb{Z}$-grading. This construction extends the construction given by Gubarev and Kolesnikov for Jordan dialgebras (arXiv:0907.1740). We introduce homotopes for Jordan pair disystems at elements and we show that we obtain again a Jordan dialgebra. Moreover, we introduce abelian inner ideals and their kernels for Leibniz algebras, and prove that the subquotient of a Leibniz algebra with respect to an abelian inner ideal is a Jordan pair disystem. This notion of subquotient extends the construction of Jordan dialgebras at $Q$-Jordan elements given by Felipe and Velásquez.
