Representation theory of quantized enveloping algebras and Hecke algebras

量化包络代数和赫克代数的表示论

• 批准号: 42722-2006
• 负责人: Bédard, Robert
• 金额: $ 0.44万
• 依托单位: Université du Québec à Montréal
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2008
• 资助国家: 加拿大
• 起止时间: 2008-01-01 至 2009-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=390499
• 关键词: Representation; theory; quantized; enveloping; algebras

Groups are algebraic structures that are related to symmetry in objects, in formulas, in structures. They appear in many context. To better understand them, one is led to realize groups concretely and  this is done through representation theory.  The representations can be broken up in simpler representations that are said to be irreducible. They are the building blocks of representation theory.    Quantized enveloping algebras and Hecke algebras are also algebraic abstract objects that generalize groups in a nice manner. They both appear in the field of Lie theory. More precisely,  Hecke algebras appear naturally when one is interested in the problem of  how to split a representation into irreducible representations for some groups, more precisely reductive groups defined over finite ou p-adic fields. The representations of these Hecke algebras are needed to analyse this splitting.  Among these, the cells representations play an essential role.  Quantized enveloping algebras (or quantum groups) also appear when one want to study  irreducible modular representation for the same groups. In both cases, the representations of these algebras can be studied with the help of canonical  bases  discovered by D. Kazhdan and G. Lusztig for Hecke algebras,  and by M.Kashiwara and G.Lusztig for quantized enveloping algebras. These bases have important properties and give good realization for the representations. However, much about them  is  still mysterious. In this research program, we will study these canonical bases and the cells representations for quantized enveloping algebras and Hecke algebras.

群是代数结构，它与对象、公式和结构中的对称性有关。它们出现在许多背景下。为了更好地理解它们，人们被引导具体地实现群，这是通过表示论来实现的。表示可以被分解为更简单的表示，这些表示被称为不可约的。它们是表征理论的基石。 量子化包络代数和Hecke代数也是代数抽象对象，以一种很好的方式推广群。它们都出现在李的理论领域。更确切地说，赫克代数自然出现时，一个感兴趣的问题是如何分裂成一个代表不可约表示的一些群体，更准确地说，还原组定义在有限欧p进领域。分析这种分裂需要Hecke代数的表示，其中胞腔表示起着至关重要的作用，当研究同一群的不可约模表示时，量子包络代数（或量子群）也会出现。在这两种情况下，这些代数的表示都可以借助于D。Kazhdan和G. Lusztig对Hecke代数的研究，以及M.Kashiwara和G.Lusztig对量子化包络代数的研究。这些基具有重要的性质，并为表示提供了很好的实现。然而，关于他们的很多事情仍然是神秘的。在这个研究计划中，我们将研究这些规范基和量子化包络代数和Hecke代数的细胞表示。