@book{c4322ac609dc42f5bbbf66861d978da4,

title = "On symmetries and cohomological invariants of equations possessing flat representations",

abstract = "We study the equation $\mathcal{E}_{\mathrm{fc}}$ of flat connections in a given fiber bundle and discover a specific geometric structure on it, which we call a \emph{flat representation}. We generalize this notion to arbitrary PDE and prove that flat representations of an equation $\mathcal{E}$ are in $1$-$1$ correspondence with morphisms $\varphi\colon\mathcal{E}\to\mathcal{E}_{\mathrm{fc}}$, where $\mathcal{E}$ and $\mathcal{E}_{\mathrm{fc}}$ are treated as submanifolds of infinite jet spaces. We show that flat representations include several known types of zero-\hspace{0pt} 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 $\varphi$ we associate a complex $C_\varphi$ of vector-\hspace{0pt} valued differential forms such that $H^1(C_\varphi)$ describes infinitesimal deformations of the flat structure, which are responsible, in particular, for parameters in B\{"}{a}cklund transformations. In addition, each higher infinitesimal symmetry $S$ of $\mathcal{E}$ defines a $1$-\hspace{0pt} cocycle $c_S$ of $C_\varphi$. Symmetries with exact $c_S$ form a subalgebra reflecting some geometric properties of $\mathcal{E}$ and $\varphi$. We show that the complex corresponding to $\mathcal{E}_{\mathrm{fc}}$ itself is $0$-\hspace{0pt} acyclic and $1$-\hspace{0pt} acyclic (independently of the bundle topology), which means that higher symmetries of $\mathcal{E}_{\mathrm{fc}}$ are exhausted by generalized gauge ones, and compute the bracket on $0$-\hspace{0pt} cochains induced by commutation of symmetries.",

keywords = "MSC-37K25, MSC-37K35, MSC-58J10, MSC-53C05, EWI-3459, IR-65826, METIS-214163",

author = "Sergei Igonine and P.H.M. Kersten and I. Krasil'shchik",

note = "Imported from MEMORANDA",

year = "2002",

language = "Undefined",

series = "memorandum",

publisher = "Fundamentele Analyse",

number = "1639",

}