Cohomology and Homotopification of averaging operators on the Lie conformal algebras

TL;DR

This paper develops the cohomology theory for averaging operators on Lie conformal algebras, introduces homotopy versions, and explores their extensions and automorphisms, advancing the algebraic understanding of these structures.

Contribution

It extends cohomology and homotopy theories to averaging Lie conformal algebras, linking 2-term $ ext{L}_ ext{infty}$-conformal algebras with cocycles and classifying extensions.

Findings

Cohomology of averaging operators on Lie conformal algebras is established.

Homotopy averaging Lie conformal algebras are introduced and connected to cocycles.

Non-abelian extensions are classified by second cohomology group.

Abstract

Building upon the work of Pavel in [P. Kolesnikov, Journal of Mathematical Physics, 56, 7 (2015)], we first present the cohomology of averaging operators on the Lie conformal algebras and use it to develop the cohomology of averaging Lie conformal algebras. We then introduce the homotopy version of averaging Lie conformal algebras and establish a connection between $2$-term averaging $\mathfrak{L}_\infty$-conformal algebra with the $3$-cocycle and crossed module of averaging Lie conformal algebra. Next, we study the non-abelian extension of the averaging Lie conformal algebras, showing that they are classified by the second non-abelian cohomology group. Finally, we demonstrate that a pair of automorphisms of averaging Lie conformal algebra is inducible if it can be seen as an image of a suitable Wells map.