Cohomology and Homotopification of averaging operators on the Lie conformal algebras

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.