Representation Theory of the Lie Algebras of Divergence Zero and Hamiltonian Vector Fields on a Torus

This thesis is about the representation theory of the Lie algebras of divergence zero and Hamiltonian vector fields on an N dimensional torus, denoted SNand HNrespectively. It combines the results of three research papers written during the doctoral work of the author. First, conditions for irreducibility of a certain class of tensor modules for SN, are investigated. Next, a classification of indecomposable and irreducible SN-modules, which have a compatible action by the commutative algebra of Laurent polynomials, and have a weight decomposition with finite dimensional weight spaces is presented. Such modules are referred to as category J, and it is seen that the tensor modules considered above comprise many of the irreducible SN-modules in category J. Similar techniques are then applied to give a classification of indecomposable and irreducible category J modules for HN.