Hom-Poisson superalgebras with nondegenerate bilinear forms and compatible Hom-anti-pre-Poisson superalgebras
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.