Do the Kontsevich tetrahedral flows preserve or destroy the space of Poisson bi-vectors?
Abstract
From the paper “Formality Conjecture” (Ascona 1996):
I am aware of only one such a class, it corresponds to simplest good graph, the complete graph with 4 vertices (and 6 edges). This class gives a remarkable vector field on the space of bi-vector fields on &R;d . The evolution with respect to the time t is described by the following non-linear partial differential equation: …, where α = ∑i,j αij∂ / ∂ xi ∧ ∂ / ∂ xj is a bi-vector field on &R;d. It follows from general properties of cohomology that 1) this evolution preserves the class of ( real-analytic ) Poisson structures , … In fact, I cheated a little bit. In the formula for the vector field on the space of bivector fields which one get from the tetrahedron graph, an additional term is present. … It is possible to prove formally that if α is a Poisson bracket, i.e. if [α, α] = 0 ∈ T2(&R; d ), then the additional term shown above vanishes . By using twelve Poisson structures with high-degree polynomial coefficients as explicit counter-examples, we show that both the above claims are false: neither does the first flow preserve the property of bi-vectors to be Poisson nor does the second flow vanish identically at Poisson bi-vectors. The counterexamples at hand suggest a correction to the formula for the “exotic” flow on the space of Poisson bi-vectors; in fact, this flow is encoded by the balanced sum involving both the Kontsevich tetrahedral graphs (that give rise to the flows mentioned above). We reveal that it is only the balance 1 : 6 for which the flow does preserve the space of Poisson bi-vectors.- Publication:
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Journal of Physics Conference Series
- Pub Date:
- January 2017
- DOI:
- 10.1088/1742-6596/804/1/012008
- arXiv:
- arXiv:1609.06677
- Bibcode:
- 2017JPhCS.804a2008B
- Keywords:
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- Mathematics - Quantum Algebra;
- Mathematical Physics;
- Mathematics - Differential Geometry
- E-Print:
- Talks given in parallel by A.B. at GADEIS VIII workshop (12--16 June 2016, Larnaca, Cyprus) and A.K. at ISQS'24 conference (13--19 June 2016, CVUT Prague, Czech Republic), 10 pages, 2 figures, 4 tables