Representation theory of quantum algebras.

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Algebra is a branch of mathematics which studies additional structures on sets that are similar to addition and multiplication of numbers. Such structured sets are also called by the generic name of "algebras". Lie algebras resemble spaces of matrices (arrays of numbers) and they are named after the late XIXe century Norwegian mathematician Sophus Lie. There is a very good team of researchers in Lie Theory spread across universities in Canada. ** I am particularly interested in algebras that are called "affine" or "double affine": for instance, matrices with entries which are polynomials in one or more variables. My work pertains also to quantum algebras and superalgebras. Very roughly speaking, quantum means replacing commutativity (xy=yx) by non-commutativity (xy is different from yx). Superalgebras are those for which a notion of even and odd elements makes sense. Affine Lie algebras, quantum algebras and superalgebras play a central role in mathematical physics (quantum mechanics, supersymmetry, etc.). ** I intend to investigate more the structure and the representations of affine, quantum and super algebras. Representations are spaces on which these mathematical objects act and they can be used to understand them better. I will use mostly algebraic methods in order to produce new representations, but geometry and combinatorics are sometimes useful. There are different forms of affine quantum algebras, some being more degenerate than others in a certain mathematical sense. One of the ideas of my research is to study first the most degenerate forms because these appear easier to understand, and then try to use the results obtained to shed light on the non-degenerate quantum algebras.