On strong n-Lie-Poisson algebras

The identities of the form ∑IcIω(xi1,…,xin)ω(xin+1,…,xi2n)=0, where ω is an antisymmetric bracket operation, are classified in the language of irreducible S2n 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.