Definition $:$ A Lie group $G$ is said to be a Poisson Lie group if it is equipped with a Poisson structure and the multiplication map $m : G \times G \longrightarrow G$ is a Poisson map i.e. for any $f, g \in C^{\infty} (G)$ and $x, y \in G$ the following equality holds $:$
$$\{f, g\} (xy) = \{f \circ \rho_{y}, g \circ \rho_{y} \} (x) + \{f \circ \lambda_{x}, g \circ \lambda_{x} \} (y)$$ where $\{\cdot, \cdot \}$ is the Poisson bracket on $G$ and $\rho_{y} : G \longrightarrow G$ and $\lambda_{x} : G \longrightarrow G$ are respectively the right translation by $y$ and the left translation by $x.$
Pavel Etingof, in his lecture notes on quantum groups, pointed out that in terms of the Poisson bivector $\Pi$ associated to $G$ the above relation takes the following form $:$ $$\Pi (xy) = (d_{x} (\rho_{y}) \otimes d_{x} (\rho_{y})) \Pi (x) + (d_{y} (\lambda_{x}) \otimes d_{y} (\lambda_{x})) \Pi (y).$$
But I can't get his point. Could anyone please shed some light on it?
Thanks for your time.
EDIT $:$ I think I have to first find out the equivalence of two definitions of bivector fields. The first one being the section of two fold wedge product of the tangent bundle and another one is given in the form of $C^{\infty} (G)$-bilinear, skew symmetric map from $\Omega^{1} (G) \times \Omega^{1} (G) \longrightarrow C^{\infty} (G).$