Leibniz homology of Lie algebras as functor homology

We prove that Leibniz homology of Lie algebras can be described as functor homology in the category of linear functors from a category associated to the Lie operad.Comment: 26 page

The functor category Fquad

26 pagesIn this paper, we define the functor category Fquad associated to vector spaces over the field with two elements, F_2, equipped with a quadratic form. We show the existence of a fully-faithful, exact functor \iota: \F \rightarrow Fquad, which preserves simple objects, where \F is the category of functors from the category of finite dimensional F_2-vector spaces to the category of all F_2-vector spaces. We define the subcategory Fiso of Fquad, which is equivalent to the product of the categories of modules over the orthogonal groups; the inclusion is a fully-faithful functor \kappa: Fiso \rightarrow Fquad which preserves simple objects

Polynomial functors from Algebras over a set-operad and non-linear Mackey functors

In this paper, we give a description of polynomial functors from (finitely generated free) groups to abelian groups in terms of non-linear Mackey functors generalizing those given in a paper of Baues-Dreckmann-Franjou-Pirashvili published in 2001. This description is a consequence of our two main results: a description of functors from (fi nitely generated free) P-algebras (for P a set-operad) to abelian groups in terms of non-linear Mackey functors and the isomorphism between polynomial functors on (finitely generated free) monoids and those on (finitely generated free) groups. Polynomial functors from (finitely generated free) P-algebras to abelian groups and from (finitely generated free) groups to abelian groups are described explicitely by their cross-e ffects and maps relating them which satisfy a list of relations.Comment: 58 page

The mixed functors of the category Fquad: a first study

In previous work, we defined the category of functors Fquad, associated to vector spaces over the field with two elements equipped with a nondegenerate quadratic form. In this paper, we define a special family of objects in the category Fquad, named the mixed functors. We give the complete decompositions of two elements of this family that give rise to two new infinite families of simple objects in the category Fquad.Comment: 24 page

Sur l'homologie des groupes d'automorphismes des groupes libres à coefficients polynomiaux

We study in this article stable homology of automorphism groups of free groups with coefficients twisted by a poynomial functor. We show that this homology is zero for a reduced covariant polynomial functor. For a reduced contravariant functor, we compute the first homology group, which is in general non zero. Our methods relie on the use of functor categories.On étudie dans cet article l'homologie stable des groupes d'automorphismes des groupes libres à coefficients tordus par un foncteur polynomial. On montre que cette homologie est nulle pour un foncteur polynomial covariant réduit. Dans le cas d'un foncteur polynomial réduit contravariant, on calcule le premier groupe d'homologie, qui n'est généralement pas nul. Nos méthodes reposent sur l'utilisation de catégories de foncteurs