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Representation theory of quantum algebras.

量子代数表示论。

基本信息

批准号：
RGPIN-2014-03589
负责人：
Guay, Nicolas
金额：
$ 1.02万
依托单位：
University of Alberta
依托单位国家：
加拿大
项目类别：
Discovery Grants Program - Individual
财政年份：
2014
资助国家：
加拿大
起止时间：
2014-01-01 至 2015-12-31
项目状态：
已结题

来源：
https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=565963
关键词：
Representation theory quantum algebras.

项目摘要

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.

代数是数学的一个分支，它研究集合上的附加结构，类似于数的加法和乘法。这样的结构集也被称为“代数”的通称。李代数类似于矩阵空间（数组），它们是以世纪挪威数学家Sophus Lie的名字命名的。有一个非常好的团队的研究人员在谎言理论分布在加拿大的大学。我特别感兴趣的代数被称为“仿射”或“双仿射”：例如，矩阵的条目是多项式在一个或多个变量。我的工作也涉及到量子代数和超代数。非常粗略地说，量子意味着用非交换性（xy不同于yx）代替交换性（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、超对称性等）中扮演着核心角色。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，但几何和组合学有时也很有用。有不同形式的仿射量子代数，有些在某种数学意义上比其他更退化。我的研究思路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阐明非简并量子代数。