Various concepts associated with quadratic algebras admit natural generalizations when the quadratic algebras are replaced by graded algebras which are finitely generated in degree 1 with homogeneous relations of degree N. Such algebras are referred to as homogeneous algebras of degree N. In particular, it is shown that the Koszul complexes of quadratic algebras generalize as N-complexes for homogeneous algebras of degree N

L'accès aux archives de la revue « Annales de l'institut Fourier » (http://annalif.ujf-grenoble.fr/) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/

We set up a homological algebra for Af-complexes, which are graded modules together with a degree -1 endomorphism d satisfying d N = 0. We define Tor-and Ext-groups for A r -complexes and we compute them m terms of their classical counterparts (N = 2) As an application, we get an alternative définition of thé Hochschild homology of an associative algebra out of an A/~complex whose differential is based on a primitive N-th root of unity. Résumé

This paper presents an analytical treatment of economic systems with an arbitrary number of agents that keeps track of the systems' interactions and agents' complexity. This formalism does not seek to aggregate agents. It rather replaces the standard optimization approach by a probabilistic description of both the entire system and agents' behaviors. This is done in two distinct steps.A first step considers an interacting system involving an arbitrary number of agents, where each agent's utility function is subject to unpredictable shocks. In such a setting, individual optimization problems need not be resolved. Each agent is described by a time-dependent probability distribution centered around his utility optimum. The entire system of agents is thus defined by a composite probability depending on time, agents' interactions and forward-looking behaviors. This dynamic system is described by a path integral formalism in an abstract space -the space of the agents' actions -and is very similar to a statistical physics or quantum mechanics system. We show that this description, applied to the space of agents' actions, reduces to the usual optimization results in simple cases.Compared to a standard optimization, such a description markedly eases the treatment of systems with small number of agents. It becomes however useless for a large number of agents. In a second step therefore, we show that for a large number of agents, the previous description is equivalent to a more compact description in terms of field theory. This yields an analytical though approximate treatment of the system. This field theory does not model the aggregation of a microeconomic system in the usual sense. It rather describes an environment of a large number of interacting agents. From this description, various phases or equilibria may be retrieved, along with individual agents' behaviors and their interactions with the environment.For illustrative purposes, this paper studies a Business Cycle model with a large number of agents.

We study the Jordan triple systems in terms of operads. We give the description of the operad of these ternary algebras as a quadratic operad and prove that the quadratic dual of this operad is the operad of partially antisymmetric, partially associative ternary algebras.Nousétudions les systèmes triples de Jordan en termes d'opérades. Nous donnons une description de l'opérade quadratique de ces algèbres ternaires et montrons que son dual quadratique est l'opérade des algèbres ternaires partiellement associatives, partiellement antisymétriques.