Irreducible weight and non-weight representations of some Lie algebras

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Lie algebra theory has become more and more widely used in many branches of mathematics and physics, including, associative algebra, algebraic geometry, topology, differential geometry, differential equation, algebraic combinatorics, string theory, conformal field theory, soliton 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 problems for the theory and its applications. For example, one important problem is how to classify irreducible representations for various Lie algebras. The main objective of my research is to study some irreducible representations for important Lie algebras from physics (such as Kac-Moddy algebras, the Virasoro algebra, and the twisted Heisenberg-Virasoro algebra), and for other Lie algebras with useful structure (for example, super-elliptic Lie algebras, quantum torus Lie algebras and Witt algebras). Solutions to these problems will benefit theoretical studies in both physics and mathematics.