Calculus of multilinear differential operators, operator $L_\infty$-algebras and ${\it IBL}_\infty$-algebras

TL;DR

This paper develops an operadic framework for describing algebraic structures with multilinear differential operators, enabling new insights into Lie and L-infinity structures and generalizing the big bracket construction.

Contribution

It introduces a multifiltration-based operadic framework for multilinear differential operators, connecting it to Lie, Lie-admissible, and L-infinity structures, and generalizing the big bracket construction.

Findings

Operadic framework for multilinear differential operators

Connection to Lie, Lie-admissible, and L-infinity structures

Generalization of the big bracket construction

Abstract

We propose an operadic framework suitable for describing algebraic structures with operations being multilinear differential operators of varying orders or, more generally, formal series of such operators. The framework is built upon the notion of a multifiltration of a linear operad generalizing the concept of a filtration of an associative algebra. We describe a particular way of constructing and analyzing multifiltrations based on a presentation of a linear operad in terms of generators and relations. In particular, that allows us to observe a special role played in this context by Lie, Lie-admissible and L-infinity structures. As a main application, and the original motivation for the present work, we show how a certain generalization of the well-known big bracket construction of Lecomte\textendash Roger and Kosmann-Schwarzbach encompassing the case of homotopy involutive Lie bialgebras can be obtained.