Research Projects

The main goal of this project is to develop the representation theory of quantum affine algebras by using their connections with the representation theory of current and affine Kac-Moody algebras. The interest in quantum affine algebras and their representations is originated from their relation to the so-called quantum Yang-Baxter equation. Finite dimensional irreducible modules provide solutions to this equation which are often referred to as R-matrices. The study of Weyl and Demazure modules - the largest indecomposable modules in this category - connects branches of mathematics from KLR algebras via PBW degenerations to combinatorics of symmetric functions, Kostka polynomials and Macdonald polynomials. The main focuses of this project are: • studying the embeddings of a given higher level affine Demazure module to a tensor product of very smaller level (mostly level one) affine Demazure modules • calculating their graded characters using these embeddings • calculating the characters of graded limits of several prime representations of quantum affine algebras using the characters of their q ›→ 1 limits As an application we expect more intriguing connections to algebraic combinatorics, such as the Schur positivity conjecture or characters of truncated Weyl modules, etc. We mainly focus on the category of finite-dimensional representations F(q) of quantum affine algebras and their connections to graded representations of current algebras. The fact that this category is not semi-simple gives a very rich structure and has many interesting consequences. However, we still have limited information on the structure of finite-dimensional irreducible representations of quantum affine algebras except for a few special cases. For example, we do not even know the dimension formulas in general. The irreducible finite-dimensional representations of quantum affine algebras were classified by Chari and Pressley around 2000 and are parametrized by tuples of Drinfeld polynomials. Two prominent approaches to the representation theory of quantum affine algebras consider particular limits of the quantum parameter: specializing the quantum parameter q to 0 or 1. Taking the q=0 limit yields the so called crystal limit of these algebras and their representations and provides combinatorial R-matrices. The q=1 limit leads to the study of representations of current algebras and is the primary focus of this project.