Some results on Hom-Novikov superalgebras and Hom-anti-pre-Novikov superalgebras

Abstract

This paper investigates the structural properties, operator representations, and algebraic duality of Hom-Novikov superalgebras and to introduce Hom-anti-pre-Novikov superalgebras. We establish a formal equivalence between Hom-Novikov superalgebras and admissible Hom-Novikov superalgebras via $q$-superalgebra extensions, proving that a Hom-pre-Lie superalgebra admits a Hom-Novikov structure if and only if its associated $2$-superalgebra is admissible. Introducing $q$-admissible pairs $(\mathcal{P},\mathcal{Q})$ on super-commutative Hom-associative superalgebras, we detail systematic constructions of Hom-Novikov superalgebras and their admissibility classes. Furthermore, we characterize anti-super-$\mathcal{O}$-operators on Hom-Lie superalgebras, showing that invertible operators induce compatible Hom-anti-pre-Lie structures and extend directly to admissible setups. Extending these frameworks to Poisson-type structures, we demonstrate that the sub-adjacent Hom-Lie superalgebra of a Hom-anti-pre-Lie Poisson superalgebra yields a transposed Hom-Poisson superalgebra. Finally, we establish the fundamental connection between Hom-anti-pre-Novikov superalgebras and invertible anti-super-$\mathcal{O}$-operators.