On symmetries and cohomological invariants of equations possessing flat representations

Sergei Igonine, P.H.M. Kersten, I. Krasil'shchik

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    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.
    Original languageUndefined
    Place of PublicationEnschede
    PublisherFundamentele Analyse
    Number of pages33
    Publication statusPublished - 2002

    Publication series

    Namememorandum
    PublisherDepartment of Applied Mathematics, University of Twente
    No.1639
    ISSN (Print)0169-2690

    Keywords

    • MSC-37K25
    • MSC-37K35
    • MSC-58J10
    • MSC-53C05
    • EWI-3459
    • IR-65826
    • METIS-214163

    Cite this

    Igonine, S., Kersten, P. H. M., & Krasil'shchik, I. (2002). On symmetries and cohomological invariants of equations possessing flat representations. (memorandum; No. 1639). Enschede: Fundamentele Analyse.