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Journal of Lie Theory 28 (2018), No. 1, 043--055
Copyright Heldermann Verlag 2018

Do n-Lie Algebras Have Universal Enveloping Algebras?

Xabier García-Martínez
Dept. of Mathematics, University of Santiago de Compostela, Lope Gomez de Marzoa, 15782 Santiago de Compostela, Spain

Rustam Turdibaev
Inha University, Ziyolilar 9, Tashkent 100170, Uzbekistan

Tim Van der Linden
Institut de Recherche en Mathématique et Physique, Université Catholique de Louvain, Chemin du cyclotron 2, 1348 Louvain-la-Neuve, Belgium


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 A. S. Dzhumadil'daev, Representations of vector product $n$-{L}ie algebras, Comm.\ Algebra 32 (2004) 3315--3326, and D. B{\u{a}}libanu and J. van de Leur, Irreducible highest weight representations of the simple $n$-Lie algebra, Transform. Groups 17 (2012) 593--613), but after analysing those we come to the conclusion that they cannot be valid in general. We give counterexamples and sufficient conditions. \par 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 {\it associated algebra functor} U: $n$-Lie$\to {\rm Alg}_\K$ inducing such an equivalence does exist, but this kind of functor never admits a right adjoint. \par We close the paper by introducing a (co)homology theory based on the associated algebra functor U.

Keywords: n-Lie, n-Leibniz, universal enveloping algebra.

MSC: 17B35

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