The heptagon-wheel cocycle in the Kontsevich graph complex
- DOI
- 10.1080/14029251.2017.1418060How to use a DOI?
- Keywords
- Non-oriented graph complex; differential; cocycle; symmetry; Poisson geometry
- 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 (2ℓ + 1)-wheel graph with a nonzero coefficient at every ℓ∈ℕ. We present detailed calculations of the differential of graphs; for the tetrahedron and pentagon-wheel cocycles, consisting at ℓ = 1 and ℓ = 2 of one and two graphs respectively, the cocycle condition d(γ) = 0 is verified by hand. For the next, heptagonwheel cocycle (known to exist at ℓ = 3), we provide an explicit representative: it consists of 46 graphs on 8 vertices and 14 edges.
- Copyright
- © 2017 The Authors. Published by Atlantis Press and Taylor & Francis
- Open Access
- This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).
Cite this article
TY - JOUR AU - Ricardo Buring AU - Arthemy V. Kiselev AU - Nina J. Rutten PY - 2021 DA - 2021/01/06 TI - The heptagon-wheel cocycle in the Kontsevich graph complex JO - Journal of Nonlinear Mathematical Physics SP - 157 EP - 173 VL - 24 IS - Supplement 1 SN - 1776-0852 UR - https://doi.org/10.1080/14029251.2017.1418060 DO - 10.1080/14029251.2017.1418060 ID - Buring2021 ER -