Transposed Novikov-Poisson algebras

TL;DR

This paper introduces transposed Novikov-Poisson algebras, explores their properties, constructions, and relations to derivations, and classifies simple cases over algebraically closed fields of characteristic zero.

Contribution

It defines transposed Novikov-Poisson algebras, studies their tensor products, constructions, and links to derivations, providing classification results for simple cases.

Findings

Tensor products of transposed Novikov-Poisson algebras are also transposed Novikov-Poisson.

Non-trivial structures exist on solvable Novikov algebras under certain conditions.

Simple transposed Novikov-Poisson algebras are one-dimensional over algebraically closed fields of characteristic zero.

Abstract

In this paper, we introduce the definition of transposed Novikov-Poisson algebras, whose affinization are transposed Poisson algebras. Moreover, we show that there is a natural transposed Poisson algebra structure on the tensor product of a transposed Novikov-Poisson algebra and a right differential Novikov-Poisson algebra. A transposed Poisson algebra also naturally arises from a transposed Novikov-Poisson algebra by taking the commutator Lie algebra of the Novikov algebra. We show that the tensor products of two transposed Novikov-Poisson algebras are also transposed Novikov-Poisson algebras. Several constructions of transposed Novikov-Poisson algebras are presented. Moreover, transposed Novikov-Poisson algebras are closely related to $\frac{1}{2}$-derivations of the associated Novikov algebras. By using $\frac{1}{2}$-derivations, we show that there are non-trivial transposed Novikov-Poisson algebra structures on solvable Novikov algebras with some conditions. We also prove that if a non-trivial transposed Novikov-Poisson algebra is simple, then the associated Novikov algebra is simple. Therefore, if the base field is algebraically closed and of characteristic 0, then any simple transposed Novikov-Poisson is of dimension $1$. Transposed Novikov-Poisson algebra structures on some simple Novikov algebras are also characterized.