Some results on anti-pre-Lie superalgebras and admissible Novikov superalgebras

In this paper, we extend the notion of anti-pre-Lie algebras to the Z(2)-graded version, and introduce the notion of anti-pre-Lie superalgebras. They can be characterized as a class of Lie-admissible superalgebras that satisfy its negative left multiplication operators are the representations of the corresponding sub-adjacent Lie superalgebras. And we give the classification of 2-dimensional anti-pre-Lie superalgebras. We introduce the notion of anti-super O-operators on Lie superalgebras to explore the relationships between anti-super O-operators and anti-pre-Lie superalgebras. We show that nondegenerate super-commutative 2-cocycles on Lie superalgebras can obtain a class of compatible anti-pre-Lie superalgebra structures. In addition, we introduce a subclass of anti-pre-Lie superalgebras, namely admissible Novikov superalgebras, which correspond to Novikov superalgebras through q-superalgebras. Finally, we introduce the notions of anti-pre-Lie Poisson superalgebras and admissible Novikov-Poisson superalgebras, extending the correspondence to the level of Poisson type structures, the correspondence of the Novikov-Poisson superalgebras and the admissible Novikov-Poisson superalgebras are realized through admissible pairs.