On $n$-pre-Lie algebras and dendrification of $n$-Lie algebras

TL;DR

This paper introduces $n$-L-dendriform algebras as a dendrification of $n$-pre-Lie algebras, explores their representations, and connects $n$-Lie algebras with phase spaces and higher pre-Lie structures.

Contribution

It defines $n$-L-dendriform algebras, studies their representations, and links $n$-Lie algebras to phase spaces and higher pre-Lie algebra constructions.

Findings

$n$-L-dendriform algebras generalize dendriform structures to $n$-ary operations.

A $n$-Lie algebra has a phase space iff it is sub-adjacent to an $n$-pre-Lie algebra.

Construction of $(n+1)$-pre-Lie algebras from $n$-pre-Lie algebras with a trace function.

Abstract

The main purpose of this paper is to introduce the notion of $n$-L-dendriform algebra which can be seen as a dendrification of $n$-pre-Lie algebras by means of $\mathcal{O}$-operators. We investigate the representation theory of $n$-pre-Lie algebras and provide some related constructions. Furthermore, we introduce the notion of phase space of a $n$-Lie algebra and show that a $n$-Lie algebra has a phase space if and only if it is sub-adjacent to a $n$-pre-Lie algebra. Moreover, we present a procedure to construct $(n + 1)$-pre-Lie algebras from $n$-pre-Lie algebras equipped with a generalized trace function.