Hom-pre-Poisson superalgebras and Dual Hom-pre-Poisson superalgebras

Abstract In this research paper, we investigate the concept of Hom-superalgebras obtained by an internal law defined on ℤ2-graded vector space A equipped with an algebra morphism f. Given a structure of Hom-pre-Lie superalgebra (A,◊,f) and Hom-Zinbiel superalgebras (A,Λ,f), we define the structure of Hom-pre-Poisson superalgebras (A,◊,Λ,f) verifying two compatibility conditions between ”◊” and ”Λ”. On the one hand, we demonstrate that when A is a Hom-pre-Lie superalgebra, then a tensoriel algebra of A has a structure of Hom-pre-Poisson superalgebra. On the other hand, we prove that any Hom-Poisson superalgebra equipped with a Baxter operator can define a structure of Hom-pre-Poisson superalgebra. Furthermore, we combine a structure of Hompermutative superalgebra and Hom-Leibniz superalgebra on the same space, we define a structure of dual Hom-pre-Poisson superalgebra and we reveal that any Hom-Poisson superalgebra equipped with an Averaging operator can define a structure of dual Hom-pre-Poisson superalgebra. (MSC) : 17A60, 17A30, 17D30, 17B63, 17A36.