Yannic
Vargas
CUNEF University Madrid
A Pre-tty Pre-amble to Pre-Lie Algebras from Combinatorial Species
Abstract.
We present an introduction to pre-Lie algebras, an algebraic structure closely related to Lie algebras. From a combinatorial perspective, pre-Lie algebras are connected to the notion of insertion of objects of a given nature into one another. In recent years, pre-Lie algebras have found numerous applications in algebra, combinatorics, quantum field theory, and numerical analysis. To better understand the nature of pre-Lie algebras, we employ the framework of species, a categorification of the concept of generating functions. This perspective allows us to describe, in a combinatorial way, several algebraic properties of pre-Lie algebras. In particular, we present a “pre-Lie-like” notion of symmetric operads, called Nested Pre-Lie operads (NPL for short). After giving several examples of NPL operads, we explain how to construct NPL-algebras, in the same way algebraic structures emerge from operads by considering algebras over operads. To do so, we use a new variant of species based on polynomial functions. This is joint work with Dominique Manchon, Hedi Regeiba, and Imen Rjaiba.