The heptagon-wheel cocycle in the Kontsevich graph complex. (21st December 2017)

• Record Type: Journal Article
• Authors: Buring, Ricardo; Kiselev, Arthemy V.; Rutten, Nina J.
• Is Part Of: Journal of nonlinear mathematical physics. Volume 24(2017)Supplement 1
• Page Start: 157
• Page End: 173
• Publication Date: 2017-12-21
• Subjects: Non-oriented graph complex -- differential -- cocycle -- symmetry -- Poisson geometry; 13D10 -- 32G81 -- 53D17 -- 81S10 -- also 53D55 -- 58J10 -- 90C35; Nonlinear theories -- Periodicals; Mathematical physics -- Periodicals; 515.252
• DOI: 10.1080/14029251.2017.1418060
• Languages: English
• ISSNs: 1402-9251

Abstract

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 2 n − 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.