Representation theory of quantum algebras.

代数是数学的一个分支，研究集合上类似于数的加法和乘法的附加结构。这种结构化的集合也被称为“代数”的总称。李代数类似于矩阵空间(数组)，它们是以十九世纪晚期挪威数学家索菲斯·李的名字命名的。在加拿大的大学里，《谎言理论》有一支非常优秀的研究团队。 我对被称为“仿射”或“双仿射”的代数特别感兴趣：例如，具有一个或多个变量多项式的矩阵。我的工作也与量子代数和超代数有关。非常粗略地说，量子意味着用非对易性(xy不同于yx)取代交换性(xy=yx)。超代数是那些用奇偶元素的概念来表示的代数。仿射李代数、量子代数和超代数在数学物理(量子力学、超对称性等)中起着核心作用。 我打算更深入地研究仿射、量子和超代数的结构和表示。表示是这些数学对象作用的空间，它们可以用来更好地理解它们。我将主要使用代数方法来产生新的表示法，但几何学和组合学有时也很有用。仿射量子代数有不同的形式，在某种数学意义上，有些比另一些更退化。我的研究想法之一是首先研究最简并的形式，因为这些形式看起来更容易理解，然后尝试使用所获得的结果来揭示非简并的量子代数。

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.