The authors consider Poisson actions of Poisson-Lie groups on symplectic manifolds. A Poisson-Lie group, a notion which was introduced by Drinfeld, is a Lie group endowed with a Poisson structure such that group multiplication is a Poisson map. J.-H. Lu defined the notion of a moment map for a Poisson action of a Poisson-Lie group, which takes values in the dual Poisson-Lie group in contrast to ordinary moment maps taking values in the dual Lie algebra. Compact Lie groups \(K\) carry a distinguished non-trivial Lie-Poisson structure known as the Lu-Weinstein structure. Alekseev showed that for every Poisson \(K\)-action on a symplectic manifold \((M,\Omega)\) with \(K^{\star}\)-valued moment map \(\Psi\), there is a different symplectic form \(\omega\) for which the action is Hamiltonian in the usual sense with a moment map \(\Phi\) taking values in the dual of the Lie algebra. Poisson reductions of \((M,\Omega,\Psi)\) are isomorphic to reductions of its linearization \((M,\omega,\Phi)\) as (stratified) symplectic spaces.
The first main result of this paper is that the linearization functor preserves operations of products, sums and conjugation on the categories of symplectic \(K\)-manifolds with both moment maps up to symplectomorphisms. The proof is based on the Moser isotopy argument. As an application the authors prove the Thomson conjecture on singular values of products of complex matrices and the corresponding statement for real matrices.
The second main result is a formula comparing the Liouville volume forms defined by \(\omega\) and \(\Omega\). This formula involves the modular function for \(K^{\star}\) and a Duflo factor. The authors obtain a hyperbolic version of Duflo isomorphism as a corollary of this result.