Journal of Lie Theory

no. 1
Do n-Lie Algebras Have Universal Enveloping Algebras?
Xabier García-Martínez1; Rustam Turdibaev2; Tim Van der Linden3
1Dept. of Mathematics, University of Santiago de Compostela, Lope Gomez de Marzoa, 15782 Santiago de Compostela, Spain
2Inha University, Ziyolilar 9, Tashkent 100170, Uzbekistan
3Institut de Recherche en Mathématique et Physique, Université Catholique de Louvain, Chemin du cyclotron 2, 1348 Louvain-la-Neuve, Belgium
Journal of Lie Theory, Volume 28 (2018) no. 1, pp. 43-55
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.
DOI: 10.5802/jolt.992
Classification: 17B35
Keywords: n-Lie, n-Leibniz, universal enveloping algebra

Xabier García-Martínez  1 ; Rustam Turdibaev  2 ; Tim Van der Linden  3

1 Dept. of Mathematics, University of Santiago de Compostela, Lope Gomez de Marzoa, 15782 Santiago de Compostela, Spain
2 Inha University, Ziyolilar 9, Tashkent 100170, Uzbekistan
3 Institut de Recherche en Mathématique et Physique, Université Catholique de Louvain, Chemin du cyclotron 2, 1348 Louvain-la-Neuve, Belgium 
Xabier García-Martínez; Rustam Turdibaev; Tim Van der Linden. Do n-Lie Algebras Have Universal Enveloping Algebras?. Journal of Lie Theory, Volume 28 (2018) no. 1, pp. 43-55. doi: 10.5802/jolt.992
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     journal = {Journal of Lie Theory},
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