The heptagon-wheel cocycle in the Kontsevich graph complex

Abstract

The real vector space of non-oriented graphs is known to carry a differential graded Lie algebra structure. Cocycles in the Kontsevich graph complex, expressed using formal sums of graphs on $n$ vertices and $2n-2$ edges, induce -- under the orientation mapping -- infinitesimal symmetries of classical Poisson structures on arbitrary finite-dimensional affine real manifolds. Willwacher has stated the existence of a nontrivial cocycle that contains the $(2\ell +1)$-wheel graph with a nonzero coefficient at every $\ell \in \mathbb{N}$. We present detailed calculations of the differential of graphs; for the tetrahedron and pentagon-wheel cocycles, consisting at $\ell =1$ and $\ell =2$ of one and two graphs respectively, the cocycle condition $d(\gamma )=0$ is verified by hand. For the next, heptagon-wheel cocycle (known to exist at $\ell =3$), we provide an explicit representative: it consists of 46 graphs on 8 vertices and 14 edges.

arXiv commentSpecial Issue JNMP 2017 `Local and nonlocal symmetries in Mathematical Physics'; 17 journal-style pages, 54 figures, 3 tables; v2 accepted

References in corpus (2)

• The Kontsevich tetrahedral flow revisited
• Do the Kontsevich tetrahedral flows preserve or destroy the space of Poisson bi-vectors?

Cited by in corpus (10)

• The calculus of multivectors on noncommutative jet spaces
• Poisson brackets symmetry from the pentagon-wheel cocycle in the graph complex
• Infinitesimal deformations of Poisson bi-vectors using the Kontsevich graph calculus
• The hidden symmetry of Kontsevich's graph flows on the spaces of Nambu-determinant Poisson brackets
• The orientation morphism: from graph cocycles to deformations of Poisson structures
• Open problems in the Kontsevich graph construction of Poisson bracket symmetries
• Kontsevich graphs act on Nambu-Poisson brackets, I. New identities for Jacobian determinants
• Universal cocycles and the graph complex action on homogeneous Poisson brackets by diffeomorphisms
• The tower of Kontsevich deformations for Nambu-Poisson structures on ${\mathbb{R}}^d$: dimension-specific micro-graph calculus
• The defining properties of the Kontsevich unoriented graph complex