Anti-pre-Lie algebras, Novikov algebras and commutative 2-cocycles on Lie algebras

TL;DR

This paper introduces anti-pre-Lie algebras as structures linked to nondegenerate commutative 2-cocycles on Lie algebras, revealing their relation to Novikov algebras and extending to Poisson algebra structures.

Contribution

It defines anti-pre-Lie algebras, explores their analogy with pre-Lie algebras, and establishes connections to Novikov and Poisson algebras, including new construction methods.

Findings

Anti-pre-Lie algebras are characterized as Lie-admissible algebras with specific representations.

Admissible Novikov algebras correspond to Novikov algebras via q-algebras.

Construction of admissible Novikov algebras from commutative algebras with derivations.

Abstract

Anti-pre-Lie algebras, Novikov algebras and commutative 2-cocycles on Lie algebrasWe introduce the notion of anti-pre-Lie algebras as the underlying algebraic structures of nondegenerate commutative 2-cocycles which are the "symmetric" version of symplectic forms on Lie algebras. They can be characterized as a class of Lie-admissible algebras whose negative left multiplication operators make representations of the commutator Lie algebras. We observe that there is a clear analogy between anti-pre-Lie algebras and pre-Lie algebras by comparing them in terms of several aspects. Furthermore, it is unexpected that a subclass of anti-pre-Lie algebras, namely admissible Novikov algebras, correspond to Novikov algebras in terms of $q$-algebras. Consequently, there is a construction of admissible Novikov algebras from commutative associative algebras with derivations or more generally, admissible pairs. The correspondence extends to the level of Poisson type structures, leading to the introduction of the notions of anti-pre-Lie Poisson algebras and admissible Novikov-Poisson algebras, whereas the latter correspond to Novikov-Poisson algebras.