Let $G$ be a [Poisson-Lie group](https://en.wikipedia.org/wiki/Poisson%E2%80%93Lie_group). Let $M$ be a symplectic manifold. 


In [the paper](https://arxiv.org/pdf/dg-ga/9602001.pdf), the third paragraph of page 1238, it is said that an action $G \times M \to M$ is called Poisson if $G \times M \to M$ is a Poisson map. 


Let $G=GL\_n$, $U$ the subgroup of $GL\_n$ consisting of all unipotent upper triangular matrices, and $T$ a maximal torus of $G$. Consider the action
$$
T \times G \to G \\
(t, g) \mapsto tgt^{-1}
$$
Is this action Poisson?


Consider the action
$$
T \times G \to G \\
(t, g) \mapsto tgw(t)^{-1}
$$
where $w \in Aut(T)$. Is this action Poisson? 


Consider the action
$$
U \times G \to G \\
(u, g) \mapsto ug
$$
Is this action Poisson?


Consider the action
$$
U \times G \to G \\
(u, g) \mapsto gu^{-1}
$$
Is this action Poisson?


Thank you very much.



Is the action $T \times G \to G$ Poisson?
Ingredients:


1. The composite of Poisson maps is Poisson
2. The action map $G\times G\to G$ is Poisson
3. Your choice of $T,U$ should be Poisson submanifolds of $G$. You didn't say which Poisson structure you're using so theoretically I can't tell, but I'm sure you're interested in the standard one in which case, they are.


Then the map $T\times G \to G\times G \to G$ is Poisson, likewise for $U$.


What's more interesting is that $T$'s Poisson structure is trivial, but $U$'s isn't, so each $\{t\}\times G\to G$ is an ichthyomorphism, but most $\{u\} \times G \to G$ aren't. 


EDIT: I misread the action. See the comment below.


