Affine and double affine quantum algebras

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Theoretical physics (conformal field theory, string theory, etc.) relies increasingly on very advanced and abstract mathematics, including analysis, geometry and algebra. My research aims to uncover new algebraic structures that are interesting from a mathematical point of view and also hold the potential for applications to questions in theoretical physics. I will investigate the representation theory of those algebraic structures. A representation is a shadow which still holds information about those structures. At the most basic level, it is a collection of matrices, which are arrays of numbers. Actually, those structures themselves can often be built using matrices, but with entries that are not just numbers but could also be, for instance, polynomials in one or several variables.      Those algebraic structures go under various names: Lie algebras (in honour of the mathematician Sophus Lie), quantum groups, quantum algebras, Yangians, quantum affine algebras, etc. They have been the subject of intense research activity by both mathematicians and physicists, for the past thirty-five years in the case of Yangians and quantum affine algebras, for over a hundred years in the case of Lie algebras. The goal of representation theory is to understand those structures better by focusing on their various representations, classifying these, decomposing them into simpler blocks, determining bases and explicit, more concrete, models. The objectives of my research program include the following: 1.Complete the classification of finite dimensional, irreducible representations of twisted Yangians of types B-C-D. Those are building blocks for larger representations. The classification should be in terms of concrete data like tuples of polynomials satisfying certain properties. 2.Construct explicit models of those representations using multi-linear algebra, combinatorics, etc. This may allow us to find bases and determine the dimension of those representations. 3.Introduce new twisted quantum affine algebras of types B,C,D using ideas similar to those used in my past work for twisted Yangians and classify also their finite dimensional, irreducible representations in terms of certain polynomials. 5. Develop the representation theory of deformed double current algebras by constructing a coproduct, studying the category O of representations and connecting it via a functor to the category of finite dimensional representations of a quantum group of finite type. Here, a category is a generalization of the notion of set and a functor is similar to a function in that it assigns one representation in one category to another one in another category, the same way that a function assigns a number to another number. 6. Construct deformed double current algebras associated to certain surfaces using ideas from quantum field theory. 7. Establish connections between affine Yangians and deformed double current algebras.