Open problems in the Kontsevich graph construction of Poisson bracket symmetries

Kiselev, Arthemy V.

doi:10.1088/1742-6596/1416/1/012018

Cited by 2 publications

(1 citation statement)

References 39 publications

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“…6 Independently, it remains an open problem (cf. [10]) whether there is a Poisson manifold (M r , P) and a graph cocycle γ such that the Poisson cohomology class of Q(P) := O r(γ)(P) would be nontrivial in H 2 P (M). In other words, for all the shifts Q = O r(γ) and all Poisson bi-vectors tried so far, the Poisson coboundary equation Q(P) = [[ X, P]] did have vector field solutions X on the manifolds M.…”

Section: Poisson Cohomology and The Graphmentioning

confidence: 99%

Universal cocycles and the graph complex action on homogeneous Poisson brackets by diffeomorphisms

Buring,

Kiselev

2019

Preprint

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The graph complex acts on the spaces of Poisson bi-vectors P by infinitesimal symmetries. We prove that whenever a Poisson structure is homogeneous, i.e. P = L V (P) w.r.t. the Lie derivative along some vector field V , but not quadratic (the coefficients of P are not degree-two homogeneous polynomials), and whenever its velocity bi-vector Ṗ = Q(P), also homogeneous w.r.t. V by L V (Q) = nQ whenever Q(P) = O r(γ)(P ⊗ n ) is obtained using the orientation morphism O r from a graph cocycle γ on n vertices and 2n − 2 edges, then the 1-vector X = O r(γ)( V ⊗ P ⊗ n−1 ) is a Poisson cocycle. Its construction is uniform for all Poisson bi-vectors P satisfying the above assumptions, on all finite-dimensional affine manifolds M . Still, if the bi-vector Q ≡ 0 is exact in the respective Poisson cohomology, so there exists a vector field Y such that Q(P) = [[ Y, P]], then the universal cocycle X does not belong to the coset of Y mod ker [[P, •]]. We illustrate the construction using two examples of cubic-coefficient Poisson brackets associated with the R-matrices for the Lie algebra gl(2).

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Section: Poisson Cohomology and The Graphmentioning

confidence: 99%

Universal cocycles and the graph complex action on homogeneous Poisson brackets by diffeomorphisms

Buring,

Kiselev

2019

Preprint

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The hidden symmetry of Kontsevich's graph flows on the spaces of Nambu-determinant Poisson brackets

Buring¹,

Lipper²,

Kiselev³

2022

Open Communications in Nonlinear Mathematical Physics

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Kontsevich's graph flows are -- universally for all finite-dimensional affine Poisson manifolds -- infinitesimal symmetries of the spaces of Poisson brackets. We show that the previously known tetrahedral flow and the recently obtained pentagon-wheel flow preserve the class of Nambu-determinant Poisson bi-vectors $P=[\![ \varrho(\boldsymbol{x})\,\partial_x\wedge\partial_y\wedge\partial_z,a]\!]$ on $\mathbb{R}^3\ni\boldsymbol{x}=(x,y,z)$ and $P=[\![ [\![\varrho(\boldsymbol{y})\,\partial_{x^1}\wedge\ldots\wedge\partial_{x^4},a_1]\!],a_2]\!]$ on $\mathbb{R}^4\ni\boldsymbol{y}$, including the general case $\varrho \not\equiv 1$. We detect that the Poisson bracket evolution $\dot{P} = Q_\gamma(P^{\otimes^{\# Vert(\gamma)}})$ is trivial in the second Poisson cohomology, $Q_\gamma = [\![ P, \vec{X}([\varrho],[a]) ]\!]$, for the Nambu-determinant bi-vectors $P(\varrho,[a])$ on $\mathbb{R}^3$. For the global Casimirs $\mathbf{a} = (a_1,\ldots,a_{d-2})$ and inverse density $\varrho$ on $\mathbb{R}^d$, we analyse the combinatorics of their evolution induced by the Kontsevich graph flows, namely $\dot{\varrho} = \dot{\varrho}([\varrho], [\mathbf{a}])$ and $\dot{\mathbf{a}} = \dot{\mathbf{a}}([\varrho],[\mathbf{a}])$ with differential-polynomial right-hand sides. Besides the anticipated collapse of these formulas by using the Civita symbols (three for the tetrahedron $\gamma_3$ and five for the pentagon-wheel graph cocycle $\gamma_5$), as dictated by the behaviour $\varrho(\mathbf{x}') = \varrho(\mathbf{x}) \cdot \det \| \partial \mathbf{x}' / \partial \mathbf{x} \|$ of the inverse density $\varrho$ under reparametrizations $\mathbf{x} \rightleftarrows \mathbf{x}'$, we discover another, so far hidden discrete symmetry in the construction of these evolution equations.

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Open problems in the Kontsevich graph construction of Poisson bracket symmetries

Cited by 2 publications

References 39 publications

Universal cocycles and the graph complex action on homogeneous Poisson brackets by diffeomorphisms

Universal cocycles and the graph complex action on homogeneous Poisson brackets by diffeomorphisms

The hidden symmetry of Kontsevich's graph flows on the spaces of Nambu-determinant Poisson brackets

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