The heptagon-wheel cocycle in the Kontsevich graph complex

Buring, Ricardo; Kiselev, Arthemy; Rutten, Nina

doi:10.48550/arxiv.1710.00658

Cited by 8 publications

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“…• k = 8: The only solution γ 7 consists of the heptagon-wheel and 45 other graphs (see Table 2, in which the coefficient of heptagon graph is 1 in bold, and [5]).…”

Section: Resultsmentioning

confidence: 99%

“…Finally, we find a heptagon-wheel eight-vertex graph which, after the orientation of its edges, gives a new universal Kontsevich flow. We refer to [8,9] for motivations, to [2,4] for an exposition of basic theory, and to [6] and [5] for more details about the pentagonwheel (5+1)-vertex and heptagon-wheel (7+1)-vertex solutions respectively. Let us remark that all the algorithms outlined here can be used without modification in the course of constructing all k-vertex Kontsevich graph solutions with higher k 5 in the deformation problem under study.Basic concept.…”

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confidence: 99%

“…2 Example 1. The sum γ 5 of two 6-vertex 10-edge graphs, γ 5 = 12 (I) ∧ 23 (II) ∧ 34 (III) ∧ 45 (IV ) ∧ 51 (V ) ∧ 16 (V I) ∧ 26 (V II) ∧ 36 (V III) ∧ 46 (IX) ∧ 56 (X) + 5 2 • 12 (I) ∧ 23 (II) ∧ 34 (III) ∧ 41 (IV ) ∧ 45 (V ) ∧ 15 (V I) ∧ 56 (V II) ∧ 36 (V III) ∧ 26 (IX) ∧ 13 (X) , is drawn in Theorem 7 on p. 10 below.…”

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confidence: 99%

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Infinitesimal deformations of Poisson bi-vectors using the Kontsevich graph calculus

Buring

Kiselev

Rutten

2018

J. Phys.: Conf. Ser.

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Let P be a Poisson structure on a finite-dimensional affine real manifold. Can P be deformed in such a way that it stays Poisson ? The language of Kontsevich graphs provides a universal approach -with respect to all affine Poisson manifolds -to finding a class of solutions to this deformation problem. For that reasoning, several types of graphs are needed. In this paper we outline the algorithms to generate those graphs. The graphs that encode deformations are classified by the number of internal vertices k; for k 4 we present all solutions of the deformation problem. For k 5, first reproducing the pentagon-wheel picture suggested at k = 6 by Kontsevich and Willwacher, we construct the heptagon-wheel cocycle that yields a new unique solution without 2-loops and tadpoles at k = 8.

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“…• k = 8: The only solution γ 7 consists of the heptagon-wheel and 45 other graphs (see Table 2, in which the coefficient of heptagon graph is 1 in bold, and [5]).…”

Section: Resultsmentioning

confidence: 99%

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confidence: 99%

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Infinitesimal deformations of Poisson bi-vectors using the Kontsevich graph calculus

Buring

Kiselev

Rutten

2018

J. Phys.: Conf. Ser.

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“…Examples of this construction for known graph cocycles γ 2ℓ+1 on n vertices and 2n − 2 edges (namely, n = 2ℓ + 2 = 4, 6, and 8) have been given in [2,3], [7], and [4], respectively. Practical calculation of graph cocycles is addressed in [8,30]; the algorithms to verify the Poisson cocycle factorisation through the Jacobi identity are available from [9]. In the fundamental work [29], see also [27], Willwacher related the unoriented graph complex to generators of the Grothendieck-Teichmüller Lie algebra grt.…”

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confidence: 99%

“…Are the graph cocycles truly graphs ? In the papers [21] (see also [22] and [4,8,28] for a pedagogical review), Kontsevich introduced the graph complex -one of the many -with parity-even vertices, with a wedge ordering of parity-odd edges, and the differential d = [•−•, •] produced by the graded commutator of graph insertions into vertices. This direction of research was furthered by Willwacher et al [11,14,29]: in particular, in [30] a generating function counts the numbers of nonzero (i.e.…”

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Open problems in the Kontsevich graph construction of Poisson bracket symmetries

Kiselev¹

2019

J. Phys.: Conf. Ser.

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Poisson brackets admit infinitesimal symmetries which are encoded using oriented graphs; this construction is due to Kontsevich (1996). We formulate several open problems about combinatorial and topological properties of the graphs involved, about integrability and analytic properties of such symmetry flows (in particular, for known classes of Poisson brackets), and about cohomological, differential geometric, and quantum aspects of the theory.

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The calculus of multivectors on noncommutative jet spaces

Kiselev

2018

Journal of Geometry and Physics

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The Leibniz rule for derivations is invariant under cyclic permutations of co-multiples within the arguments of derivations. We explore the implications of this principle: in effect, we construct a class of noncommutative bundles in which the sheaves of algebras of walks along a tesselated affine manifold form the base, whereas the fibres are free associative algebras or, at a later stage, such algebras quotients over the linear relation of equivalence under cyclic shifts. The calculus of variations is developed on the infinite jet spaces over such noncommutative bundles.In the frames of such field-theoretic extension of the Kontsevich formal noncommutative symplectic (super)geometry, we prove the main properties of the Batalin-Vilkovisky Laplacian and Schouten bracket. We show as by-product that the structures which arise in the classical variational Poisson geometry of infinite-dimensional integrable systems do actually not refer to the graded commutativity assumption.

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The heptagon-wheel cocycle in the Kontsevich graph complex

Cited by 8 publications

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Infinitesimal deformations of Poisson bi-vectors using the Kontsevich graph calculus

Infinitesimal deformations of Poisson bi-vectors using the Kontsevich graph calculus

Open problems in the Kontsevich graph construction of Poisson bracket symmetries

The calculus of multivectors on noncommutative jet spaces

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