AI 中文总结
本文引入Hom-anti-Zinbiel超代数等概念，通过非退化超交换上循环建立Hom-Poisson超代数与相容Hom-anti-pre-Poisson超代数的联系，为两类超代数提供统一研究框架。
AI 中文摘要
本文首先引入Hom-anti-Zinbiel超代数的概念，将对应的反代数结构扩展到Hom超情形，发展其基本性质并通过anti-super-$\boldsymbol{\rm O}$-算子和anti-Rota-Baxter算子对其进行刻画。此外，我们引入超交换Hom结合超代数上的超交换Connes上循环，以及Hom-Lie超代数上的超交换2-上循环的概念。我们证明，非退化超交换Connes上循环会产生相容的Hom-anti-Zinbiel超代数结构，而非退化超交换2-上循环则诱导出相容的Hom-anti-pre-Lie超代数结构。作为主要应用，我们证明：配备了其Hom结合分量上的非退化超交换Connes上循环、且其Hom-Lie分量上配备了非退化超交换2-上循环的Hom-Poisson超代数，可典范地确定一个相容的Hom-anti-pre-Poisson超代数，反之亦然。这提供了一个连接Hom-Poisson超代数与Hom-anti-pre-Poisson超代数的统一框架，并揭示了非退化超对称双线性型在Hom型超代数结构理论中的基础作用，同时还获得了若干进一步的结构结果与刻画。
英文摘要
In this paper, we first introduce the notions of Hom-anti-Zinbiel superalgebras, extending the corresponding anti-algebraic structures to the Hom-super setting. We develop their fundamental properties and characterize them in terms of anti-super-$\mathcal O$-operators and anti-Rota-Baxter operators. Furthermore, we introduce the concepts of super-commutative Connes cocycles on super-commutative Hom-associative superalgebras and super-commutative $2$-cocycles on Hom-Lie superalgebras. We prove that nondegenerate super-commutative Connes cocycles give rise to compatible Hom-anti-Zinbiel superalgebra structures, while nondegenerate super-commutative $2$-cocycles induce compatible Hom-anti-pre-Lie superalgebra structures. As a main application, we show that a Hom-Poisson superalgebra equipped with a nondegenerate super-commutative Connes cocycle on its Hom-associative component and a nondegenerate super-commutative $2$-cocycle on its Hom-Lie component canonically determines a compatible Hom-anti-pre-Poisson superalgebra, and conversely. This provides a unified framework linking Hom-Poisson and Hom-anti-pre-Poisson superalgebras and reveals the fundamental role played by nondegenerate supersymmetric bilinear forms in the structure theory of Hom-type superalgebras. Several further structural results and characterizations are also obtained.
