This paper aims to introduce the notion of anti-pre-Lie algebras and then study the relationships between them and the related structures such as anti-\(\mathcal O\)-operators, commutative 2-cocycles on Lie algebras, and Novikov algebras. Let us give some important definitions.
Definition. A vector space \(A\) with a bilinear operation is called a pre-Lie algebra if the following equation is satisfied: Namely, an algebra \(A\) is a pre-Lie algebra if its associator is left-symmetric, i.e. \((x,y,z)=(y,x,z),\) where \((x,y,z)=(xy)z-x(yz)\). It is known that each pre-Lie algebra is Lie-admissible, i.e. under the commutator product \([x,y]=xy-yz\) it gives a Lie algebra. Using an analogy between associative and anti-associative algebras, the author define the notion of anti-associator \((x,y,z)_{aa}:=(xy)z+x(yz).\) The class of anti-pre-Lie algebras is defined as a class of Lie-admissible algebras with the left-symmetric anti-associator. Namely, we have the following definition.
Definition. A vector space \(A\) with a bilinear operation is called an anti-pre-Lie algebra if the following equations are satisfied:
Section 2 is dedicated to the study of anti-pre-Lie algebras and their properties. Namely,

1.

it was proved that each commutative anti-pre-Lie algebra is associative [Proposition 2.6];

2.

some examples of anti-pre-Lie algebras of lengths one are given in Proposition 2.7;

3.

a classification of complex \(2\)-dimensional anti-pre-Lie algebras is given in Proposition 2.10.

Then they introduce the notion of anti-\(\mathcal O\)-operators [Definition 2.12] to interpret anti-pre-Lie algebras. There is a close relationship between anti-pre-Lie algebras and commutative 2-cocycles on Lie algebras. That is, the former can be induced from the latter in the nondegenerate case, whereas there is a natural construction of the latter on the semi-direct product Lie algebras induced from the former. Finally, they summarize these results to exhibit a clear analogy between anti-pre-Lie algebras and pre-Lie algebras.
In Section 3, they first introduce the notion of admissible Novikov algebras as a subclass of anti-pre-Lie algebras, corresponding to Novikov algebras in terms of \(q\)-algebras.
Definition. A Novikov algebra is a pre-Lie algebra \(A\) such that
Definition. An admissible Novikov algebra is an algebra \(A\) such that:
It is easy to see that an admissible Novikov algebra is an anti-pre-Lie algebra [Proposition 3.3.]. Then they give the constructions of Novikov algebras and the corresponding admissible Novikov algebras from commutative associative algebras with admissible pairs including derivations. Finally, they extend the correspondence between Novikov algebras and admissible Novikov algebras to the level of Poisson-type structures and hence introduce the notions of anti-pre-Lie Poisson algebras and admissible Novikov-Poisson algebras [Definition 3.35]. The relationships with transposed Poisson algebras and Novikov-Poisson algebras [Proposition 3.46] as well as a tensor theory are given [Theorem 3.48].

Reviewer: Ivan Kaygorodov (Covilhã)

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