Representation theory of quantum algebras.

代数是数学的一个分支，它研究集合上的附加结构，类似于数字的加法和乘法。这样的结构化集合也被称为“代数”。李代数类似于矩阵空间（数字数组），它们是以19世纪晚期挪威数学家索菲斯·李命名的。在加拿大的大学里，有一支非常优秀的李氏理论研究团队。我对被称为“仿射”或“双仿射”的代数特别感兴趣：例如，矩阵的项是一个或多个变量的多项式。我的工作也涉及量子代数和超代数。非常粗略地说，量子意味着用非交换性（xy不同于yx）取代交换性（xy=yx）。超代数是那些偶数和奇数元素的概念是有意义的。仿射李代数、量子代数和超代数在数学物理（量子力学、超对称等）中起着核心作用。我打算进一步研究仿射代数、量子代数和超代数的结构和表示。表征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。为了生成新的表示，我将主要使用代数方法，但几何和组合有时也很有用。有不同形式的仿射量子代数，在某种数学意义上，有些比其他的更简并。我研究的一个想法是首先研究最简并的形式，因为它们看起来更容易理解，然后尝试使用获得的结果来阐明非简并的量子代数。

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.