Infinitesimal bialgebras, pre-Lie and dendriform algebras

TL;DR

This paper explores the structure of infinitesimal bialgebras, introducing related categories and showing how to derive pre-Lie and dendriform algebras, with applications to quivers and endomorphisms, and extends these concepts to bimodules.

Contribution

It introduces categories of modules over infinitesimal bialgebras and constructs new algebraic structures like pre-Lie, dendriform, and brace algebras from them.

Findings

Pre-Lie algebra can be constructed from any infinitesimal bialgebra.

A dendriform structure arises on endomorphisms of an infinitesimal bialgebra.

A pre-Lie structure is found on the space of paths on any quiver.

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.