Beyond Kirillov--Reshetikhin modules: character formulae and highest weight categories

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The main theme of the proposal is to study connections between the representation theory of affine Lie algebras and their quantum analogs and other algebraic structures. The theory of finite dimensional representations of affine algebras shows many of the same properties as modular representation theory and is remarkably complex. The PI intends to show that one can associate quasi-hereditary algebras to such categories and to explore the possibility of proving an analog of the famous BGG duality for these representations. Many of these ideas will also work in the greater generality of extended affine Lie algebras. Another theme of the project is to understand the relationship between representations of quantum affine algebras and cluster algebras. An important concept in this is the idea of prime representations and minimal affinizations and the PI plans to develop a deeper understanding of these ideas. Affine Lie algebras and their quantum analogs have remarkable connections to a number of different fields including string theory, conformal field theory, topological field theory, infinite dimensional geometry and mathematical physics. Important physical phenomena can be better explained and predicted via mathematical theories built on them. Representations of affine Lie algebras and their quantum analogs will capture important physical information and are the core of the building the above connections to other fields. The main theme of this project is to study the representation theory of affine Lie algebras and their quantum analogs.