On symmetries and cohomological invariants of equations possessing flat representations

We study the equation epsilon(fc) of flat connections in a given fiber bundle and discover a specific geometric structure on it, which we call a flat representation. We generalize this notion to arbitrary PDE and prove that flat representations of an equation epsilon are in 1-1 correspondence with morphisms rho: epsilon-->epsilon(fc), epsilon and epsilon(fc) being treated as submanifolds of infinite jet spaces. We show that flat representations include several known types of zero-curvature formulations of PDEs. In particular, the Lax pairs of the self-dual Yang-Mills equations and their reductions are of this type. With each flat representation rho we associate a complex C-rho of vector-valued differential forms such that H-1 (C-rho) describes infinitesimal deformations of the flat structure, which are responsible, in particular, for parameters in Backlund transformations. In addition, each higher infinitesimal symmetry S of epsilon defines a 1-cocycle c(S) of C-rho. Symmetries with exact c(S) form a subalgebra reflecting some geometric properties of epsilon and rho. We show that the complex corresponding to epsilon(fc) itself is 0-acyclic and I-acyclic (independently of the bundle topology), which means that higher symmetries of epsilon(fc) are exhausted by generalized gauge ones, and compute the bracket on 0-cochains induced by commutation of symmetries. (C) 2003 Elsevier B.V. All rights reserved.