The Kontsevich graph orientation morphism revisited

Abstract

The orientation morphism $Or(\cdot )(P):\gamma \mapsto \dot{P}$ 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 $\mathbf{\gamma }\in \ker d$ 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 $\Pi T^{\ast }{N}^d$ 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 $\Pi T^{\ast }{N}^d$.

Int. workshop on Homotopy algebras, deformation theory and quantization (16-22 September 2018, Bedlewo, Poland), to appear in Banach Center Publications; 5 figures, 18 pages

References in corpus (2)

• Open problems in the Kontsevich graph construction of Poisson bracket symmetries
• Formality morphism as the mechanism of $\star$-product associativity: how it works

Cited by in corpus (4)

• The hidden symmetry of Kontsevich's graph flows on the spaces of Nambu-determinant Poisson brackets
• Open problems in the Kontsevich graph construction of Poisson bracket symmetries
• Kontsevich graphs act on Nambu--Poisson brackets, II. The tetrahedral flow is a coboundary in 4D
• Universal cocycles and the graph complex action on homogeneous Poisson brackets by diffeomorphisms