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

Question

Let $G$ be a Poisson-Lie group. Let $M$ be a symplectic manifold.

In the paper, 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.

Answer 1

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.