Academic literature on the topic 'Chevalley-Eilenberg'

Le but de ce travail est d'expliquer en quoi l'application de d'antisymétrisation de Cartan-Eilenberg F*, qui permet d'identifer la cohomologie de Chevalley-Eilenberg d'une algèbre de Lie g à la cohomologie de Hochschild de son algèbre enveloppante Ug, est l'analogue algébrique de l'application usuelle de dérivation de cochaînes de groupe lisses au voisinage de l'élément neutre d'un groupe de Lie, et comment un de ses quasi-inverses peut être construit et compris comme une application d'intégration de cocycles de Lie. De plus, nous montrons qu'un tel quasi-inverse, bien que provenant d'une contraction d'origine géométrique, peut s'écrire de manière totalement intinsèque, en n'utilisant que la structure d'algèbre de Hopf cocommutative connexe sur Ug<br>This thesis aims at explaining why Cartan and Eilenberg's antisymmetrisation map F*, which provides an explicit identifcation between the Chevalley-Eilenberg cohomology of a free lie algebra g and the Hochschild cohomology of its universal enveloping algebra Ug, can be seen as an algebraic analogue of the well-known derivation map from the complex of locally smooth group cochains to the one of Lie algebra cochains, and how one of its quasi-inverses can be built and thought of as an integration of Lie algebra cochains in Lie group cochains process. Moreover, we show that such a quasi-inverse, even if it is defined thanks to a Poincare contraction coming from geometry, can be written using a totally intrinsic formula that involves only the connex cocommutative Hopf algebra structure on Ug

L’obiettivo principale di questa tesi è lo studio dei complessi misti graduati e del loro possibile ruolo nell’ambito della ricerca nella teoria della deformazione. In particolare, mi sono occupato della relazione tra complessi misti graduati e algebre di Lie derivate. I due contributi principali di questa tesi sono la costruzione di una t-struttura completa e non-degenere sulla ∞-categoria stabile dei complessi misti graduati, che esibisce i moduli misti graduati come il completamento sinistro della t-struttura di Beilinson sulla ∞-categoria derivata filtrata, e la costruzione di una famiglia di funtori di Chevalley-Eilenberg verso la ∞-categoria dei moduli misti graduati. Nonostante sia noto che i complessi di Chevalley-Eilenberg siano dotati di questa struttura, in letteratura tale funtore non è mai stato costruito in maniera indipendente da qualsiasi modello per le ∞-categorie.<br>The main objective of this thesis is to study mixed graded complexes as a framework where to study derived deformation theory. In particular, I investigated the relationship between mixed graded complexes and derived Lie algebras. The main contributions of this thesis are the following. First, I provided the ∞-category of mixed graded complexes with a complete and non-degenerate t-structure, which exhibits such ∞-category as the left completion of the Beilinson t-structure on the filtered derived ∞-category. Secondly, I constructed a family of Chevalley-Eilenberg ∞-functors computing homology and cohomology of derived Lie algebras, endowing them of a richer structure of mixed graded complexes. Even if it is known that Chevalley-Eilenberg complexes are endowed with such structure, my construction is new and completely model-independent.