Weight representations of lie algebras

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Lie algebra theory has become more and more widely used in many branches of mathematics, physics, and finance. Associative algebra theory, topology, differential geometry, algebraic combinatorics, theory of 2-dimensional statistical models, string theory, conformal field theory, soliton theory, and other mathematical physics all use Lie algebra theory. Besides being useful in many subjects, Lie algebra theory is inherently attractive, combining a great depth and a satisfying degree of completeness in its basic theory. It also presents many interesting and important problems. For example, how to classify various representations for different Lie algebras and find out their character formulas. The main objective of my research is to study good representations for some important Lie algebras from physics (such as Kac-Moddy algebras, generalized Virasoro algebras and the twisted Heisenberg-Virasoro algebra), and for some other Lie algebras with useful structure (for example, quantum torus Lie algebras and generalized Cartan type Lie algebras). Solutions to these problems will advance theoretical studies in both physics and mathematics.