Dual coalgebras of Jacobian $n$-Lie algebras over polynomial rings

We establish the structure of the dual Lie coalgebra for a Lie algebra of the symplectic Poisson bracket (Jacobian-type Poisson bracket) on the algebra of polynomials in evenly many variables. We show that if the base field has characteristic zero then the $n$-ary dual coalgebra for the Jacobian $n$-Lie algebra consists of the same linear functionals as the dual coalgebra for the commutative polynomial algebra.