The Kontsevich graph orientation morphism revisited
Tom 123 / 2021
Streszczenie
The orientation morphism 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.