Summary: INFINITESIMAL BIALGEBRAS, PRE-LIE AND DENDRIFORM ALGEBRAS
Abstract. We introduce the categories of infinitesimal Hopf modules and bimodules over an infin-
itesimal 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.
The main results of this paper establish connections between infinitesimal bialgebras, pre-Lie alge-
bras and dendriform algebras, which were a priori unexpected.
An infinitesimal bialgebra (abbreviated -bialgebra) is a triple (A, µ, ) where (A, µ) is an associative
algebra, (A, ) is a coassociative coalgebra, and is a derivation (see Section 2). We write (a) =
a1a2, omitting the sum symbol.
Infinitesimal bialgebras were introduced by Joni and Rota [17, Section XII]. The basic theory of