On strong n-Lie–Poisson algebras

The identities of the form image, where ω is an antisymmetric bracket operation, are classified in the language of irreducible image modules. This classification implies that the notions of strong n-Lie–Poisson and n-Lie–Poisson algebras coincide if and only if n is odd. This is the answer to a problem of Dzhumadil’daev.