A bi-Hamiltonian manifold is a manifold $M$ equipped with two compatible Poisson tensors $\pi_0$ and $\pi$.

I am interested in the case of a Lie group $G$ endowed with a multiplicatif Poisson tensor $\pi$ and a left invariant Poisson tensor $\pi_0$ (associated to a left invariant symplectic form), such that the two tensors are compatible, i.e. $[\pi,\pi_0]=0$.

I wonder if the dual Lie group $G^\star$ is also of the same kind, i.e. it admits a left invariant symplectic form $\omega^\star$ such that the associated Poisson tensor $\pi_0^\star$ is compatible with the dual multiplicatif Poisson tensor $\pi^\star$.

I am doing some bibliographic research to see if such problem was already been studied.