The space T poly (R d ) of all tensor fields on R d , equipped with the Schouten bracket is a Lie algebra. The subspace of ascending tensors is a Lie subalgebra of T poly (R d ).In this paper, we compute the cohomology of the adjoint representations of this algebra (in itself and T poly (R d )), when we restrict ourselves to cochains defined by aerial Kontsevitch's graphs like in our previous work (Pacific J of Math, vol 229, no 2, (2007) 257-292). As in the vectorial graphs case, the cohomology is freely generated by all the products of odd wheels.
Résumé.L'espace des tenseurs ascendants est une sous algèbre de Lie de l'algèbre de Lie (graduée) T poly (R d ) des champs de tenseurs sur R d muni du crochet de Schouten.Dans cet article, on calcule la cohomologie des représentations adjointes de cette sous algèbre de Lie, en se restreignant à des cochaînes définies par des graphes de Kontsevich aériens comme dans [AAC1] et [AAC2]. On retrouve un résultat analogue à celui de la cohomologie des graphes vectoriels et linéaires: elle est librement engendrée par des produits de roues de longueurs impaires.
Let T poly (R d ) denote the space of skew-symmetric polyvector fields on R d , turned into a graded Lie algebra by means of the Schouten bracket. Our aim is to explore the cohomology of this Lie algebra, with coefficients in the adjoint representation, arising from cochains defined by linear combination of aerial Kontsevich graphs. We prove that this cohomology is localized at the space of graphs without any isolated vertex, any "hand" or any "foot". As an application, we explicitly compute the cohomology of the "ascending graphs" quotient complex.
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