Do $n$-Lie algebras have universal enveloping algebras

TL;DR

This paper investigates the existence and properties of universal enveloping algebras for $n$-Lie algebras, showing they generally do not exist and exploring related categorical and homological structures.

Contribution

It demonstrates the non-existence of universal enveloping algebras for $n$-Lie algebras in general and introduces a related (co)homology theory based on an associated algebra functor.

Findings

Counterexamples show universal enveloping algebras generally do not exist for $n$-Lie algebras.

An associated algebra functor exists but lacks a right adjoint.

A new (co)homology theory for $n$-Lie algebras is proposed.

Abstract

The aim of this paper is to investigate in which sense, for $n\geq 3$, $n$-Lie algebras admit universal enveloping algebras. There have been some attempts at a construction (see [10] and [5]) but after analysing those we come to the conclusion that they cannot be valid in general. We give counterexamples and sufficient conditions. We then study the problem in its full generality, showing that universality is incompatible with the wish that the category of modules over a given $n$-Lie algebra $L$ is equivalent to the category of modules over the associated algebra $U(L)$. Indeed, an associated algebra functor $U \colon \text{$n$-}\mathsf{Lie}_{\mathbb{K}} \to \mathsf{Alg}_{\mathbb{K}}$ inducing such an equivalence does exist, but this kind of functor never admits a right adjoint. We end the paper by introducing a (co)homology theory based on the associated algebra functor $U$.