The orientation morphism: From graph cocycles to deformations of Poisson structures

Ricardo Buring*, Arthemy Kiselev

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review

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We recall the construction of the Kontsevich graph orientation morphism $\gamma \mapsto {\rm O\vec{r}}(\gamma)$ which maps cocycles $\gamma$ in the non-oriented graph complex to infinitesimal symmetries $\dot{\mathcal{P}} = {\rm O\vec{r}}(\gamma)(\mathcal{P})$ of Poisson bi-vectors on affine manifolds. We reveal in particular why there always exists a factorization of the Poisson cocycle condition $[\![\mathcal{P},{\rm O\vec{r}}(\gamma)(\mathcal{P})]\!] \doteq 0$ through the differential consequences of the Jacobi identity $[\![\mathcal{P},\mathcal{P}]\!]=0$ for Poisson bi-vectors $\mathcal{P}$. To illustrate the reasoning, we use the Kontsevich tetrahedral flow $\dot{\mathcal{P}} = {\rm O\vec{r}}(\gamma_3)(\mathcal{P})$, as well as the flow produced from the Kontsevich--Willwacher pentagon-wheel cocycle $\gamma_5$ and the new flow obtained from the heptagon-wheel cocycle $\gamma_7$ in the unoriented graph complex.
Original languageEnglish
Article number012017
Number of pages10
JournalJournal of Physics: Conference Series
Publication statusPublished - 19-Nov-2018
Event The 32nd International Colloquium on Group Theoretical Methods in Physics (Group32) - Prague, Czech Republic
Duration: 9-Jul-201813-Jul-2018


  • math.CO
  • math.DG
  • math.QA
  • math.SG
  • 05C22, 68R10, 16E45, 53D17, 81R60

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