The project addresses questions in the representation theory of affine Lie algebras and their quantum analogs. The problems to be studied are motivated by applications to mathematical physics and combinatorics. One of the goals of the project is to make a rigorous connection between combinatorial results on crystals and the corresponding results on representations of the affine Lie algebra. This involves studying the category of finite dimensional representations of the affine algebra, the closely related current algebra and the quantum analogs of these algebras for generic values of the quantum parameter. This category of representations is not semisimple and exhibits features similar to modular representation theory. This is an important motivation for the study of Weyl modules and of extensions between irreducible representations of these algebras undertaken by the project. The study of Weyl modules and the conjecture on their dimension made by Chari and Pressley is related to the conjectures of Feigin and Loktev on the fusion product of finite dimensional irreducible representations of a simple Lie algebra coming from their study on conformal field theory . It is expected that the results on the Weyl modules and their quotients the Kirrillov/Reshetikhin modules will provide further insight into and also lead to generalizations of the conjectures and constructions of Feigin and Loktev. An important open problem is to determine a character formula for the irreducible finite dimensional representations of the quantum affine algebras analogous to the Weyl character formula. A first step is to study this problem for the Kirrillov/Reshetikhin modules and the project pursues this by seeing if a conjecture of Dorey made in his study of affine Toda field theories is correct for these modules.
The representation theory of affine Lie algebras and the quantum algebras is an area where there is intense research activity. It has had significant impact on other branches of mathematics such as number theory, knot theory, combinatorics to name a few. It has had fruitful interaction with mathematical physics, in affine Toda field theories, and in solvable models in statistical mechanics where the tensor product of representations of affine algebras conjecturally describes the interactions or fusing of particles. The PI's study should confirm some of these theories and also have applications in the representation theory of affine Lie algebras of positive level and vertex algebra constructions of such representations.
