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Modules for the Lie algebra of Hamiltonian vector fields on a torus, whichadmit a compatible action for the commutative algebra of multivariate Laurentpolynomials are called category $\mathcal{J}$. This paper classifies theindecomposable and the irreducible modules in category $\mathcal{J}$.

Category $\mathcal{J}$ Modules for Hamiltonian Vector Fields on a Torus