Reference : An explicit two step quantization of Poisson structures and Lie bialgebras |

Scientific journals : Article | |||

Physical, chemical, mathematical & earth Sciences : Mathematics | |||

http://hdl.handle.net/10993/28968 | |||

An explicit two step quantization of Poisson structures and Lie bialgebras | |

English | |

Merkulov, Sergei [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >] | |

Willwacher, Thomas [ETH, Zurich > Mathematics] | |

2018 | |

Communications in Mathematical Physics | |

Springer | |

364 | |

2 | |

505–578 | |

Yes (verified by ORBi^{lu}) | |

0010-3616 | |

1432-0916 | |

Germany | |

[en] Quantization ; Poisson structures ; Lie bialgebras | |

[en] We develop a new approach to deformation quantizations of Lie bialgebras and Poisson structures which goes in two steps.
In the first step one associates to any Poisson (resp. Lie bialgebra) structure a so called quantizable Poisson (resp. Lie bialgebra) structure. We show explicit transcendental formulae for this correspondence. In the second step one deformation quantizes a quantizable Poisson (resp. Lie bialgebra) structure. We show again explicit transcendental formulae for this second step correspondence (as a byproduct we obtain configuration space models for biassociahedron and bipermutohedron). In the Poisson case the first step is the most non-trivial one and requires a choice of an associator while the second step quantization is essentially unique, it is independent of a choice of an associator and can be done by a trivial induction. We conjecture that similar statements hold true in the case of Lie bialgebras. The main new result is a surprisingly simple explicit universal formula (which uses only smooth differential forms) for universal quantizations of finite-dimensional Lie bialgebras. | |

http://hdl.handle.net/10993/28968 | |

57 pages |

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