Trying to understand dressing actions

Question

I am reading the lecture notes and trying to understand dressing actions.

Let $G$ be a Poisson-Lie group and $G^*$ its dual Poisson-Lie group. In the lecture notes above, Proposition 5.22 on page 80, it is said that there is a dressing action $Dr: G^* \times G \to G$ given by $Dr(g_*)(g) = h$, where $h \in G$ such that there is $h_*$ such that $g_* \cdot g = h \cdot h_*$. I am trying to understand this formula. Is "$\cdot$" the usual product of elements in a group? For example, let $G=SL_2$ and $g \in G$. How to compute the $h$ in $Dr(g_*)(g) = h$? Thank you very much.

Answer 1

I think that first you should understand that if your Poisson-Lie group has zero Poisson structure then the dressing action of $G$ on $G^*\simeq \mathfrak g^*$ is just the usual coadjoint action.

Let $G=SU(2)$ and consider the so-called standard Poisson-Lie group structure. This implies $D=SL(2,\mathbb C)$ and $$ G^*\simeq \left\{\begin{pmatrix}a&b\\ 0&a^{-1} \end{pmatrix} : a\in\mathbb R_{>0}, b\in\mathbb C\right\}\, . $$ The fact that you can write both: $$ D=SU(2)G^*=G^*SU(2) $$ is just an instance of Iwasawa decomposition and a simple computation with $2\times 2$ matrices will give you quite explicit formulae. If you wish to control results you may find them here: Ahluwalia.

More generally the Iwasawa dcomposition, written as both a left and right Iwasawa decomposition, gives you the factorizations needed to write down in principle explicit formulae for the dressing action. I say in principle because in practice writing down explicit formulae for a given matrix in $SL(n;\mathbb C)$ may be quite difficult.

Of course usually you're not that interested in explicit formulae, rather in properties of this action e.g. a description of its orbits which coincide with the symplectic foliation that for a compact standard Poisson-Lie group is by now well understood.