Tangent lifts of bi-Hamiltonian structures

We construct two Poisson structures πTM and π̃TM on the tangent bundle TM to a Poisson manifold (M,π) using Lie algebroid (T*M, qM, M). Next, we construct the Poisson manifold (TM,πTM,λ) from a bi-Hamiltonian manifold (M,π1,π2). This structure can be considered as a deformation of the Poisson structure related to an algebroid structure. We show that the bi-Hamiltonian structure from M can be transferred to the manifold TM. Moreover we present how to construct the Casimir functions for structures πTM, πTM,λ, π̃Tg*, and π̃Tg*,λ from the Casimir functions for π1 and π2 and discuss some particular examples.