Goldman–Turaev formality implies Kashiwara–Vergne

  • Anton Alekseev

    Université de Genève, Switzerland
  • Nariya Kawazumi

    University of Tokyo, Japan
  • Yusuke Kuno

    Tsuda University, Tokyo, Japan
  • Florian Naef

    Trinity College, Dublin, Ireland
Goldman–Turaev formality implies Kashiwara–Vergne cover
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Let be a compact connected oriented 2-dimensional manifold with non-empty boundary. In our previous work, we have shown that every solution of the higher genus Kashiwara–Vergne equations for an automorphism of a free Lie algebra induces an isomorphism between the Goldman–Turaev Lie bialgebra and its associated graded gr . In this paper, we prove the converse: if induces an isomorphism , then it satisfies the Kashiwara–Vergne equations up to conjugation. As an application of our results, we compute the degree one non-commutative Poisson cohomology of the Kirillov–Kostant–Souriau double bracket. The main technical tool used in the paper is a novel characterization of conjugacy classes in the free Lie algebra in terms of cyclic words.

Cite this article

Anton Alekseev, Nariya Kawazumi, Yusuke Kuno, Florian Naef, Goldman–Turaev formality implies Kashiwara–Vergne. Quantum Topol. 11 (2020), no. 4, pp. 657–689

DOI 10.4171/QT/143