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Algèbres de Lie d'homotopie associées à une proto-bigèbre de Lie

Published online by Cambridge University Press:  20 November 2018

Momo Bangoura
Momo Bangoura*
Affiliation:
Département de Mathématiques, Université de Conakry, BP 1147, République de Guinée email: [email protected]
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Résumé

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On associe à toute structure de proto-bigèbre de Lie sur un espace vectoriel $F$ de dimension finie des structures d’algèbre de Lie d’homotopie définies respectivement sur la suspension de l’algèbre extérieure de $F$ et celle de son dual ${{F}^{*}}$. Dans ces algèbres, tous les crochets $n$-aires sont nuls pour $n\,\ge \,4$ du fait qu’ils proviennent d’une structure de proto-bigèbre de Lie. Plus généralement, on associe à un élément de degré impair de l’algèbre extérieure de la somme directe de $F$ et ${{F}^{*}}$, une collection d’applications multilinéaires antisymétriques sur l’algèbre extérieure de $F$ (resp. ${{F}^{*}}$), qui vérifient les identités de Jacobi généralisées, définissant les algèbres de Lie d’homotopie, si l’élément donné est de carré nul pour le grand crochet de l’algèbre extérieure de la somme directe de $F$ et de ${{F}^{*}}$.

Abstract

Abstract

To any proto-Lie algebra structure on a finite-dimensional vector space $F$, we associate homotopy Lie algebra structures defined on the suspension of the exterior algebra of $F$ and that of its dual ${{F}^{*}}$, respectively. In these algebras, all $n$-ary brackets for $n\,\ge \,4$ vanish because the brackets are defined by the proto-Lie algebra structure. More generally, to any element of odd degree in the exterior algebra of the direct sum of $F$ and ${{F}^{*}}$, we associate a set of multilinear skew-symmetric mappings on the suspension of the exterior algebra of $F$ (resp. ${{F}^{*}}$), which satisfy the generalized Jacobi identities, defining the homotopy Lie algebras, if the given element is of square zero with respect to the big bracket of the exterior algebra of the direct sum of $F$ and ${{F}^{*}}$.

Keywords

17B7017A30algèbre de Lie d'homotopiebigèbre de Liequasi-bigèbre de Lieproto-bigèbre de Liecrochet dérivéjacobiateur
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 59 , Issue 4 , 01 August 2007 , pp. 696 - 711
Copyright
Copyright © Canadian Mathematical Society 2007

References

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