ABSTRACT
Let G- https://www.w3.org/1998/Math/MathML"> P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072393/02402e9d-0cfd-4d5e-bccf-46dd4134d466/content/eq1415.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> be a compact simple Poisson-Lie group equipped with a Poisson structure https://www.w3.org/1998/Math/MathML"> P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072393/02402e9d-0cfd-4d5e-bccf-46dd4134d466/content/eq1416.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and (M, ω) be a symplectic manifold. Assume that M carries a Poisson action of G https://www.w3.org/1998/Math/MathML"> P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072393/02402e9d-0cfd-4d5e-bccf-46dd4134d466/content/eq1417.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and there is an equivariant moment map in the sense of Lu and Weinstein which maps to the dual Poisson-Lie group https://www.w3.org/1998/Math/MathML"> G P * , m : M → G P * https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072393/02402e9d-0cfd-4d5e-bccf-46dd4134d466/content/eq1418.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . M can always be equipped with another symplectic form https://www.w3.org/1998/Math/MathML"> ω ˜ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072393/02402e9d-0cfd-4d5e-bccf-46dd4134d466/content/eq1419.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> so that the G-action preserves https://www.w3.org/1998/Math/MathML"> ω ˜ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072393/02402e9d-0cfd-4d5e-bccf-46dd4134d466/content/eq1420.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and there is a new moment map https://www.w3.org/1998/Math/MathML"> μ = e − 1 ∘ m : M → g * https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072393/02402e9d-0cfd-4d5e-bccf-46dd4134d466/content/eq1421.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> to the dual space of the Lie algebra. Here e is a universal invertible equivariant map https://www.w3.org/1998/Math/MathML"> e : g * → G P * https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072393/02402e9d-0cfd-4d5e-bccf-46dd4134d466/content/eq1422.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . We suggest an alternative proof of the convexity theorem for the Poisson-Lie moment map recently proved by Flaschka and Ratiu [1].
