Bruhat Poisson structure on CPn and integrable systems

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In this note we present a simple formula for the Bruhat Poisson structure on complex projective spaces in terms of the momentum coordinates. We also give a simple description of a family of functions in involution on compact Hermitian symmetric spaces obtained via the bi-Hamiltonian approach using the Bruhat Poisson structure and an invariant symplectic structure. We compute these functions explicitly on CPn and relate them to the Gelfand–Tsetlin coordinates. We also show how the Lenard scheme can be applied.