# The Kontsevich graph orientation morphism revisited

## TL;DR

This paper revisits the Kontsevich graph orientation morphism, providing a new formulation that expresses it directly in terms of graphs without supermathematical tools, enhancing understanding of Poisson structures.

## Contribution

It offers a reformulation of the orientation morphism formula directly in terms of graphs, removing the need for supermathematical constructs.

## Key findings

- Reformulation of the orientation morphism in graph terms.
- Connection between graph cocycles and Poisson cocycles.
- Enhanced understanding of graph-based Poisson geometry.

## Abstract

The orientation morphism $Or(\cdot)(P)\colon \gamma\mapsto\dot{P}$ associates differential-polynomial flows $\dot{P}=Q(P)$ on spaces of bi-vectors $P$ 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 $d$-cocycles $\boldsymbol{\gamma}\in\ker d$ with respect to the vertex-expanding differential $d=[{\bullet}\!\!{-}\!{-}\!\!{\bullet},\cdot]$ are mapped by $Or$ to Poisson cocycles $Q(P)\in\ker\,[\![ P,{\cdot}]\!]$, that is, to infinitesimal symmetries of Poisson bi-vectors $P$. The formula of orientation morphism $Or$ 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 $\Pi T^* N^d$.