On the transposed Poisson n-Lie algebras

TL;DR

This paper explores the structure of transposed Poisson n-Lie algebras, establishing conditions for their simplicity and analyzing their properties, including the failure of strong conditions in free cases.

Contribution

It generalizes simplicity criteria for transposed Poisson n-Lie algebras and examines their structural properties under specific compatibility conditions.

Findings

Transposed Poisson n-Lie algebras are strong under certain conditions.

A transposed Poisson n-Lie algebra is simple iff its associated n-Lie algebra is simple.

The strong condition fails for free transposed Poisson 3-Lie algebras.

Abstract

We study unital commutative associative algebras and their associated n-Lie algebras, showing that they are strong transposed Poisson n-Lie algebras under specific compatibility conditions. Furthermore, we generalize the simplicity criterion for transposed Poisson algebras, proving that a transposed Poisson n-Lie algebra is simple if and only if its associated n-Lie algebra is simple. In addition, we study the strong condition for transposed Poisson n-Lie algebras, proving that it fails in the case of a free transposed Poisson 3-Lie algebra.