Representation theory of quantum algebras.

量子代数表示论。

• 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: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=587298
• 关键词: Representation; theory; quantum; algebras.

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，它研究了与数字的添加和乘法相似的集合的其他结构。这种结构化集也以“代数”的通用名称调用。谎言代数类似于物品的空间（数字阵列），它们以十九世纪后期挪威数学家索菲斯（Sophus）谎言命名。有一个非常好的研究人员在谎言理论中分布在加拿大的大学中。 我对称为“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中多项式的条目的物品。我的工作也涉及量子代数和超级法。从非常粗略的角度来看，量子意味着通过非交换性替换通勤性（xy = yx）（xy与yx不同）。超级堡是一个偶数和奇怪元素的概念的那些。仿射为代数，量子代数和超级别在数学物理学（量子力学，超对称性等）中起着核心作用。 我打算更多地研究仿射，量子和超级代数的结构和表示。表示是这些数学对象作用的空间，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形式，但是几何和组合有时很有用。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代数更退化。我的研究的思想之一是首先研究最简单的形式，因为这些形式似乎更容易理解，然后尝试将获得的结果阐明，以阐明非排分量子代数。