Infinitesimal bialgebras, pre-Lie and dendriform algebras

Abstract

We introduce the categories of infinitesimal Hopf modules and bimodules over an infinitesimal bialgebra. We show that they correspond to modules and bimodules over the infinitesimal version of the double. We show that there is a natural, but non-obvious way to construct a pre-Lie algebra from an arbitrary infinitesimal bialgebra and a dendriform algebra from a quasitriangular infinitesimal bialgebra. As consequences, we obtain a pre-Lie structure on the space of paths on an arbitrary quiver, and a striking dendriform structure on the space of endomorphisms of an arbitrary infinitesimal bialgebra, which combines the convolution and composition products. We extend the previous constructions to the categories of Hopf, pre-Lie and dendriform bimodules. We construct a brace algebra structure from an arbitrary infinitesimal bialgebra; this refines the pre-Lie algebra construction. In two appendices, we show that infinitesimal bialgebras are comonoid objects in a certain monoidal category and discuss a related construction for counital infinitesimal bialgebras.