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H 1 Poisson-Lie duality and Langlands duality via Bohr-Sommerfeld

Auteurs : Alekseev, Anton (Auteur de la Conférence)
CIRM (Editeur )

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    Résumé : Let $G$ be a connected semisimple Lie group with Lie algebra $\mathfrak{g}$. There are two natural duality constructions that assign to it the Langlands dual group $G^\lor$ (associated to the dual root system) and the Poisson-Lie dual group $G^∗$. Cartan subalgebras of $\mathfrak{g}^\lor$ and $\mathfrak{g}^∗$ are isomorphic to each other, but $G^\lor$ is semisimple while $G^∗$ is solvable.
    In this talk, we explain the following non-trivial relation between these two dualities: the integral cone defined by the Berenstein-Kazhdan potential on the Borel subgroup $B^\lor \subset G^\lor$ is isomorphic to the integral Bohr-Sommerfeld cone defined by the Poisson structure on $K^∗ \subset G^∗$ (the Poisson-Lie dual of the compact form $K \subset G$). The first cone parametrizes canonical bases of irreducible $G$-modules. The corresponding points in the second cone belong to integral symplectic leaves of $K^∗$.
    The talk is based on a joint work with A. Berenstein, B. Hoffman and Y. Li.

    Keywords : Langlands dual; Poisson-Lie duality; cluster algebras; potentials; tropicalization

    Codes MSC :
    17B10 - Representations of Lie algebras, algebraic theory
    53D17 - Poisson manifolds; Poisson groupoids and algebroids

      Informations sur la Vidéo

      Langue : Anglais
      Date de publication : 22/10/2018
      Date de captation : 10/10/2018
      Sous collection : Research talks
      Format : MP4
      arXiv category : Representation Theory ; Spectral Theory
      Domaine : Topology ; Lie Theory and Generalizations ; Geometry
      Durée : 00:53:56
      Audience : Chercheurs ; Doctorants , Post - Doctorants
      Download : https://videos.cirm-math.fr/2018-10-10_Alekseev.mp4

    Informations sur la rencontre

    Nom de la rencontre : International workshop on geometric quantization and applications / Colloque international "Quantification géométrique et applications"
    Organisateurs de la rencontre : Ma, Xiaonan ; Meinrenken, Eckhard ; Paradan, Paul-Emile
    Dates : 08/10/2018 - 12/10/2018
    Année de la rencontre : 2018
    URL Congrès : https://conferences.cirm-math.fr/1867.html

    Citation Data

    DOI : 10.24350/CIRM.V.19464603
    Cite this video as: Alekseev, Anton (2018). Poisson-Lie duality and Langlands duality via Bohr-Sommerfeld. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19464603
    URI : http://dx.doi.org/10.24350/CIRM.V.19464603


    Voir aussi

    Bibliographie

    1. Alekseev, A., Berenstein, A., Hoffman, B., & Li, Y. (2018). Langlands duality and Poisson-Lie duality via cluster theory and tropicalization. - https://arxiv.org/abs/1806.04104

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