On the operad of associative algebras with derivation

TL;DR

This paper explores the operad structure of associative algebras with derivation, linking it to polynomial and rational function substitution, and introduces homotopy variants within this framework.

Contribution

It characterizes the operad of associative algebras with derivation via polynomials and rational functions, connecting to moulds and formal group laws, and extends to homotopy associative algebras.

Findings

Operad determined by polynomials in several variables and substitution.

Rational functions lead to an operad isomorphic to moulds.

Introduces homotopy associative algebra with derivation.

Abstract

We study the operad of associative algebras equipped with a derivation. We show that it is determined by polynomials in several variables and substitution. Replacing polynomials by rational functions gives an operad which is isomorphic to the operad of "moulds". It provides an efficient environment for doing integro-differential calculus. Interesting variations are obtained by using formal group laws. The preceding case corresponds to the additive formal group law. We unravel the notion of homotopy associative algebra with derivation in the spirit of Kadeishvili's work.