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A combinatorial approach to coefficients in deformation quantization
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Graph cocycles for star-products are investigated from the combinatorial point of view, using Connes-Kreimer renormalization techniques. The Hochschild complex, controlling the deformation theory of associative algebras, is the ``Kontsevich representation'' of a DGLA of graphs coming from a pre-Lie algebra structure defined by graph insertions. Properties of the dual of its UEA (an odd parity analog of Connes-Kreimer Hopf algebra), are investigated in order to find solutions of the deformation equation. The solution of the initial value deformation problem, at tree-level, is unique. For linear coefficients the resulting formulas are relevant to the Hausdorff series.
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Cited by 1 Pith paper
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From Kontsevich Graphs to Feynman graphs, a Viewpoint from the Star Products of Scalar Fields
A Moyal-like product on functions is lifted to scalar fields and functionals, and its graph expansion is shown to reproduce the known adjacency-matrix description of Feynman graphs.
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