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A local normal form for Hamiltonian actions of compact semisimple Poisson–Lie groups

Megumi Harada, Jeremy Lane and Aidan Patterson

Vol. 15 (2022), No. 5, 775–812

The main contribution of this manuscript is a local normal form for Hamiltonian actions of Poisson–Lie groups K on a symplectic manifold equipped with an AN-valued moment map, where AN is the dual Poisson–Lie group of K. Our proof uses the delinearization theorem of Alekseev which relates a classical Hamiltonian action of K with a 𝔨-valued moment map to a Hamiltonian action with an AN-valued moment map, via a deformation of symplectic structures. We obtain our main result by proving a “delinearization commutes with symplectic quotients” theorem which is also of independent interest and then putting this together with the local normal form theorem for classical Hamiltonian actions with ^˛-valued moment maps. A key ingredient for our main result is the delinearization 𝒟(ωcan ) of the canonical symplectic structure on TK, so we additionally take some steps toward explicit computations of 𝒟(ωcan ). In particular, in the case K = SU (2), we obtain explicit formulas for the matrix coefficients of 𝒟(ωcan ) with respect to a natural choice of coordinates on TSU (2).

Hamiltonian actions, moment maps, Poisson–Lie groups, local normal forms
Mathematical Subject Classification
Primary: 53D20
Secondary: 53D17
Received: 1 September 2021
Revised: 9 February 2022
Accepted: 20 February 2022
Published: 3 March 2023

Communicated by Michael Jablonski
Megumi Harada
Department of Mathematics and Statistics
McMaster University
Hamilton, ON
Jeremy Lane
Department of Mathematics
McMaster University
Hamilton, ON
Aidan Patterson
Department of Mathematics
University of Waterloo
Waterloo, ON