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Library for manipulation of Kontsevich graphs and sums and series of them (in C++11).
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kontsevich_graph.cpp
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kontsevich_graph_operator.hpp
kontsevich_graph_series.hpp
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kontsevich_graph_sum.tpp
kontsevich_graph_weight.hpp
leibniz_graph.hpp
leibniz_graph.tpp

README.md

kontsevich_graph_series-cpp

Library for manipulation of Kontsevich graphs and sums and series of them (in C++11).

General features:

  • generate Kontsevich graphs,
  • manipulate sums and series of graphs with arbitrary (e.g. numeric, symbolic) coefficients,
  • reduce sums and series of graphs modulo skew-symmetry,
  • reduce graph series modulo the Jacobi identity and its differential consequences,
  • evaluate graph series at particular Poisson structures,
  • for a graph series that should vanish for all Poisson structures, obtain relations between the coefficients by substituting particular Poisson structures,
  • substitute relations between coefficients back into a graph series,
  • write the polydifferential operator associated to a graph series as a LaTeX formula.

Star product features:

  • construct a star product from a list of admissible graphs and (possibly undetermined) weights,
  • obtain cyclic weight relations for graphs with undetermined weights,
  • compute the associator (f★g)★h - f★(g★h) for a graph star product,
  • obtain weight relations from the associativity constraint for particular Poisson structures,
  • gauge-transform a star product,
  • invert a gauge transformation,
  • calculate the weight integrand of a graph as a rational function of Cartesian coordinates.

Poisson cohomology features:

  • skew-symmetrize graph series,
  • calculate the Schouten bracket of polyvectorfields given by skew Kontsevich graph series,
  • reduce graph polyvectorfields modulo coboundaries [[P, X]] where X is a graph series.

Dependencies:

  • GiNaC and its dependency CLN, for:
    • kontsevich_graph_operator.hpp,
    • kontsevich_graph_weight.hpp,
    • util/continued_fraction.hpp,
    • util/poisson_structure.hpp,
    • util/poisson_structure_examples.hpp,
    • and the test programs.
  • Eigen, for linear algebra in
    • tests/reduce_mod_jacobi.cpp,
    • tests/reduce_mod_jacobi_iterative.cpp.
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