Differential Operator Endomorphisms of an Euler-Lagrange Complex

Abstract

The main results of our paper deal with the lifting problem for multilinear differential operators between complexes of horizontal de Rham forms on the infinite jet bundle. We answer the question when does an n-multilinear differential operator from the space of (N,0)-forms (where N is the dimension of the base) to the space of (N-s,0)-forms allow an n-multilinear extension of degree (-s,0) defined on the whole horizontal de Rham complex. To study this problem we define a differential graded operad DEnd of multilinear differential endomorphisms, which we prove to be acyclic in positive degrees (negative mapping degrees) and describe the cohomology group in degree zero in terms of the characteristic. As a corollary, we solve the lifting problem. An important application to mathematical physics is the proof of existence of a strongly homotopy Lie algebra structure extending a Lie bracket on the space of functionals.