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The Kontsevich graph orientation morphism revisited

Volume 123 / 2021

Arthemy V. Kiselev, Ricardo Buring Banach Center Publications 123 (2021), 123-139 MSC: Primary 05C22, 16E45, 53D17; Secondary 68R10, 81R60. DOI: 10.4064/bc123-5

Abstract

The orientation morphism ${\sf O}\vec{{\sf r}}(\cdot)({\cal P})\colon\gamma\mapsto\dot{{\cal P}}$ associates differential-polynomial flows $\dot{{\cal P}}={\cal Q}({\cal P})$ on spaces of bi-vectors ${\cal P}$ on finite-dimensional affine manifolds $N^d$ with (sums of) finite unoriented graphs $\gamma$ with ordered sets of edges and without multiple edges and one-cycles. It is known that ${\rm d}$-cocycles $\boldsymbol{\gamma}\in\ker{\rm d}$ with respect to the vertex-expanding differential ${\rm d}=[{\bullet}\!\!{-}\!{-}\!\!{\bullet},\cdot]$ are mapped by $\mathsf{O}\vec{\mathsf r}$ to Poisson cocycles ${\cal Q}({\cal P})\in\ker\,[\![{\cal P},{\cdot}]\!]$, that is, to infinitesimal symmetries of Poisson bi-vectors ${\cal P}$. The formula of orientation morphism $\mathsf{O}\vec{\mathsf r}$ was expressed in terms of the edge orderings as well as parity-odd and parity-even derivations on the odd cotangent bundle $\Pi T^* N^d$ over any $d$-dimensional affine real Poisson manifold $N^d$. We express this formula in terms of (un)oriented graphs themselves, i.e. without explicit reference to supermathematics on $\Pi T^* N^d$.

Authors

  • Arthemy V. KiselevBernoulli Institute for Mathematics
    Computer Science and Artificial Intelligence
    University of Groningen
    P.O.Box 407
    9700 AK Groningen, The Netherlands
    e-mail
  • Ricardo BuringInstitut für Mathematik
    Johannes Gutenberg-Universität
    Staudingerweg 9
    D-55128 Mainz, Germany
    e-mail

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