A Dixmier theorem for Poisson enveloping algebras

We consider a skew-symmetric n -ary bracket on the polynomial algebra K [ x 1 , … + ] ( ≥ 2 ) over field of characteristic zero defined by { } = Jac C where is fixed element and Jacobian. If then this Poisson if 3 it an -Lie-Poisson . describe center corresponding show that quotient / − λ ideal generated 0 ≠ ∈ simple central homogeneous not proper power any nonzero polynomial. This construction includes quotients P sl enveloping Lie standard Casimir in It also proven M exceptional seven dimensional Malcev are simple,