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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
##### 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 $AN$-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 ${\mathfrak{𝔨}}^{\ast }$-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 $^˛\ast$-valued moment maps. A key ingredient for our main result is the delinearization $\mathsc{𝒟}\left({\omega }_{\mathrm{can}}\right)$ of the canonical symplectic structure on ${T}^{\ast }\phantom{\rule{-0.17em}{0ex}}K$, so we additionally take some steps toward explicit computations of $\mathsc{𝒟}\left({\omega }_{\mathrm{can}}\right)$. In particular, in the case $K=\mathrm{SU}\left(2\right)$, we obtain explicit formulas for the matrix coefficients of $\mathsc{𝒟}\left({\omega }_{\mathrm{can}}\right)$ with respect to a natural choice of coordinates on ${T}^{\ast }\phantom{\rule{-0.17em}{0ex}}\mathrm{SU}\left(2\right)$.

##### Keywords
Hamiltonian actions, moment maps, Poisson–Lie groups, local normal forms
Primary: 53D20
Secondary: 53D17