Simple smooth representations of 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, geometric representation, vertex operator algebra, partial differential equations, 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. There are still many interesting problems for the theory and its applications. For example, one important problem is how to classify different classes of irreducible representations for various Lie algebras. Representation theory for Lie algebras is far from being well developed. One reason is that it is generally considered impossible to classify all irreducible modules over any nontrivial Lie algebras as pointed out by Dixmier. So far, only the 2-dimensional non-abelian Lie algebra, the 3-dimensional simple Lie algebra, the 3-dimensional Heisenberg algebra and some of their deformations have such classifications. The main objective of my research is to study some irreducible representations for important Lie algebras in physics, such as affine Kac-Moody algebras, the Virasoro algebra, the twisted Heisenberg-Virasoro algebra, and other Lie algebras with useful structure and applications (for example, super-elliptic Lie algebras and Witt algebras).   Another objective of my program is to apply Lie algebra theory to the study of isomorphisms and automorphisms of algebraic curves in algebraic geometry, and to the study of automorphisms of polynomial rings, Aut(C[x1, x2, ..., xn]), which is a long standing problem for n>2 in commutative algebra. More precisely, in the next five years, I plan to carry out the following investigations. 1. Classify irreducible smooth representations for affine Kac-Moody algebras, in particular, irreducible restricted representations for affine Kac-Moody algebras; 2. Classify irreducible weight representations with finite dimensional weight spaces for Witt algebras W+n , and determine higher height (>1) simple smooth modules over W+n ; 3. Construct new simple weight or non-weight sln+1-modules and characterize some classes of known simple sln+1-modules; 4. Use simple representations of Heisenberg algebras to construct new simple weight or non-weight modules over the finite dimensional Lie algebras of type Cn and G2; 5. Develop Krichever-Novikov algebra theory of higher genus to study isomorphisms and automorphisms of algebraic curves in algebraic geometry. Solutions to these problems will benefit studies in both physics and mathematics.