0000000000154617
AUTHOR
Ridha Chatbouri
Algèbres et cogèbres de Gerstenhaber et cohomologie de Chevalley–Harrison
Resume Un prototype des algebres de Gerstenhaber est l'espace T poly ( R d ) des champs de tenseurs sur R d muni du produit exterieur et du crochet de Schouten. Dans cet article, on decrit explicitement la structure de la G ∞ algebre enveloppante d'une algebre de Gerstenhaber. Cette structure permet de definir une cohomologie de Chevalley–Harrison sur cette algebre. On montre que cette cohomologie a valeur dans R n'est pas triviale dans le cas de la sous algebre de Gerstenhaber des tenseurs homogenes T poly hom ( R d ) .
Hom-Lie quadratic and Pinczon Algebras
ABSTRACTPresenting the structure equation of a hom-Lie algebra 𝔤, as the vanishing of the self commutator of a coderivation of some associative comultiplication, we define up to homotopy hom-Lie algebras, which yields the general hom-Lie algebra cohomology with value in a module. If the hom-Lie algebra is quadratic, using the Pinczon bracket on skew symmetric multilinear forms on 𝔤, we express this theory in the space of forms. If the hom-Lie algebra is symmetric, it is possible to associate to each module a quadratic hom-Lie algebra and describe the cohomology with value in the module.
Chevalley cohomology for aerial Kontsevich graphs
Let $T_{\operatorname{poly}}(\mathbb{R}^d)$ denote the space of skew-symmetric polyvector fields on $\mathbb{R}^d$, 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.