Some Problems On The Heisenberg-Virasoro Algebra

The Heisenberg-Virasoro algebra is an important infinite-dimensional Lie algebra,which has important applications in mathematics and physics. In this dissertation, we stud-ied some problems on the structure and representation theory of the Heisenberg-Virasoro algebra. In the first part, we investigated the structures of a class of modules on which the positive part of the algebra acts locally-finitely, based on some known results on this topics. For some special case, we can give a description of the modules. In the second part,we studied the Rota-Baxter operators of Heisenberg-Virasoro algebra, and in particular,we give a characterization of the homogeneous Rota-Baxter operators of weight 0.