Homology with coefficients of Leibniz <i>n</i>-algebras

José Manuel Casas

doi:10.1016/j.crma.2009.04.004

Homological Algebra

Homology with coefficients of Leibniz n-algebras

Presented by: Christophe Soulé

José Manuel Casas ¹

¹Dpto. Matemática Aplicada I, Univ. de Vigo, 36005 Pontevedra, Spain

Comptes Rendus. Mathématique, Volume 347 (2009) no. 11-12, pp. 595-598

Abstract (Original language)
Résumé

Co-representations of Leibniz n-algebras are defined as left modules over the universal enveloping algebra. We define the homology of a Leibniz n-algebra L with coefficients in a co-representation M as the homology of the Leibniz complex of $L^{\otimes n - 1}$ over the co-representation $M \otimes L$ .

We prove the cancellation of the homology over free objects and the generalization of the following isomorphism in Leibniz homology $H L_{⋆} (L, L) ≅ H L_{⋆ + 1} (L, K)$ from Leibniz algebras to Leibniz n-algebras.

Les co-représentations des n-algèbres de Leibniz sont définies comme les modules à gauche sur l'algèbre enveloppante universelle. Nous définissons l'homologie de la n-algèbre de Leibniz L à coefficients dans une co-représentation M comme l'homologie du complexe de Leibniz de $L^{\otimes n - 1}$ sur la co-représentation $M \otimes L$ .

Nous démontrons l'annulation de l'homologie sur les objets libres et nous généralisons l'isomorphisme $H L_{⋆} (L, L) ≅ H L_{⋆ + 1} (L, K)$ des algèbres de Leibniz aux n-algèbres de Leibniz.

Received: 2008-06-11
Accepted: 2009-04-02
Published online: 2009-04-21

DOI: 10.1016/j.crma.2009.04.004

Author's affiliations:

José Manuel Casas ¹

¹ Dpto. Matemática Aplicada I, Univ. de Vigo, 36005 Pontevedra, Spain

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     title = {Homology with coefficients of {Leibniz} \protect\emph{n}-algebras},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {595--598},
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     doi = {10.1016/j.crma.2009.04.004},
     language = {en},
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José Manuel Casas. Homology with coefficients of Leibniz n-algebras. Comptes Rendus. Mathématique, Volume 347 (2009) no. 11-12, pp. 595-598. doi: 10.1016/j.crma.2009.04.004

References
Cited by

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