Commutators of pre Lie $n$-algebras and $P{L}_{\infty }$-algebras

Abstract

We show that a $P{L}_{\infty }$-algebra $V$ can be described by a nilpotent coderivation of degree $-1$ on coalgebra $P^{\ast }V$. Based on this result, we can generalise the result of T. Lada and show that every $A_{\infty }$-algebra carries a $P{L}_{\infty }$-algebra structure and every $P{L}_{\infty }$-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.