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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
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Abstract |
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The main contribution of this manuscript is a local normal form for Hamiltonian actions of Poisson–Lie groups on a symplectic manifold equipped with an -valued moment map, where is the dual Poisson–Lie group of . Our proof uses the delinearization theorem of Alekseev which relates a classical Hamiltonian action of with a -valued moment map to a Hamiltonian action with 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 of the canonical symplectic structure on , so we additionally take some steps toward explicit computations of . In particular, in the case , we obtain explicit formulas for the matrix coefficients of with respect to a natural choice of coordinates on . |
Keywords
Hamiltonian actions, moment maps, Poisson–Lie groups, local
normal forms
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Mathematical Subject Classification
Primary: 53D20
Secondary: 53D17
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Milestones
Received: 1 September 2021
Revised: 9 February 2022
Accepted: 20 February 2022
Published: 3 March 2023
Communicated by Michael Jablonski
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Authors |
| Megumi Harada | |
| Department of Mathematics and
Statistics McMaster University Hamilton, ON Canada |
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| Jeremy Lane | |
| Department of Mathematics McMaster University Hamilton, ON Canada |
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| Aidan Patterson | |
| Department of Mathematics University of Waterloo Waterloo, ON Canada |
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