Commutators of pre Lie $n$-algebras and $PL_{\infty}$-algebras

TL;DR

This paper explores the relationships between different algebraic structures, showing how $PL_{ infty}$-algebras relate to $A_{ infty}$ and $L_{ infty}$-algebras, and establishing new structures on partially associative $n$-algebras.

Contribution

It generalizes existing results by connecting $PL_{ infty}$-algebras with $A_{ infty}$ and $L_{ infty}$-algebras and introduces pre Lie $n$-algebra structures on partially associative $n$-algebras.

Findings

$PL_{ infty}$-algebras can be described via nilpotent coderivations.

Every $A_{ infty}$-algebra admits a $PL_{ infty}$-structure.

Partially associative $n$-algebras have induced pre Lie $n$-algebra structures.

Abstract

We show that a $PL_{\infty}$-algebra $V$ can be described by a nilpotent coderivation of degree $-1$ on coalgebra $P^*V$. Based on this result, we can generalise the result of T. Lada and show that every $A_{\infty}$-algebra carries a $PL_{\infty}$-algebra structure and every $PL_{\infty}$-algebra carries an $L_{\infty}$-algebra structure. In particular, we obtain a pre Lie $n$-algebra structure on an arbitrary partially associative $n$-algebra and deduce pre Lie $n$-algebras are $n$-Lie admissible.