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

Harada, Megumi; Lane, Jeremy; Patterson, Aidan

doi:10.2140/involve.2022.15.775

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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

DOI: 10.2140/involve.2022.15.775

Abstract

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 $A N$ -valued moment map, where $A N$ 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 $A N$ -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 $𝒟 (ω_{can})$ of the canonical symplectic structure on $T^{*} K$ , so we additionally take some steps toward explicit computations of $𝒟 (ω_{can})$ . In particular, in the case $K = SU (2)$ , we obtain explicit formulas for the matrix coefficients of $𝒟 (ω_{can})$ with respect to a natural choice of coordinates on $T^{*} SU (2)$ .

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Keywords

Hamiltonian actions, moment maps, Poisson–Lie groups, local normal forms

Mathematical Subject Classification

Primary: 53D20

Secondary: 53D17

Milestones

Received: 1 September 2021

Revised: 9 February 2022

Accepted: 20 February 2022

Published: 3 March 2023

Communicated by Michael Jablonski

Authors

Megumi Harada
Department of Mathematics and Statistics McMaster University Hamilton, ON Canada

Jeremy Lane
Department of Mathematics McMaster University Hamilton, ON Canada

Aidan Patterson
Department of Mathematics University of Waterloo Waterloo, ON Canada

Vol. 15, No. 5, 2022