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

TL;DR

This paper establishes a local normal form for Hamiltonian actions of compact semisimple Poisson-Lie groups on symplectic manifolds with an $AN$-valued moment map, extending classical results through delinearization techniques.

Contribution

It introduces a novel local normal form for Hamiltonian Poisson-Lie group actions using delinearization and symplectic quotient compatibility, with explicit computations for $SU(2)$.

Findings

Derived explicit formulas for delinearized symplectic structures on $T^*SU(2)$.

Proved delinearization commutes with symplectic quotients.

Extended classical Hamiltonian normal forms to Poisson-Lie group actions.

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 $\mathfrak{k}^*$-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 wtih $\mathfrak{k}^*$-valued moment maps. A key ingredient for our main result is the delinearization $\mathcal{D}(\omega_{can})$ of the canonical symplectic structure on $T^*K$, so we additionally take some steps toward explicit computations of $\mathcal{D}(\omega_{can})$. In particular, in the case $K=SU(2)$, we obtain explicit formulas for the matrix coefficients of $\mathcal{D}(\omega_{can})$ with respect to a natural choice of coordinates on $T^*SU(2)$.