Some new homological invariants of nonlinear differential equations. I (Q1902714)

Using the Frölicher-Nijenhuis bracket as the basic algebraic device to define curvature, it is shown by construction that to a given nonlinear partial differential equation, \(\varepsilon\) say, there is a naturally associated complex. The scheme is quite general and relevant examples are provided. The motivation for doing such a construction is of course the general experience that there is much insight to be gained from the study of the corresponding cohomology. Indeed, analysis here shows that the modules \(H^0 (\varepsilon)\) and \(H^1 (\varepsilon)\) are naturally identified to the Lie algebra of higher infinitesimal symmetries of \(\varepsilon\) and the equivalence classes of infinitesimal deformations of \(\varepsilon\), respectively. Spectral sequences and relevant applications are to appear in a later issue of the same journal.