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: 2006
• 资助国家: 加拿大
• 起止时间: 2006-01-01 至 2007-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=339643
• 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.

群是与对象、公式、结构中的对称性相关的代数结构。它们出现在许多背景下。为了更好地理解它们，人们被引导去具体地认识群，这是通过表征理论来完成的。它们是表示论的积木，量子化的包络代数和Hecke代数也是用很好的方式泛化群的代数抽象对象。它们都出现在谎言理论领域。更准确地说，当人们对如何将表示分裂成某些群的不可约表示感兴趣时，Hecke代数自然地出现了，更准确地说，是定义在有限ou p-ady域上的约化群。要分析这种分裂，需要这些Hecke代数的表示。在这些表示中，胞表示起着至关重要的作用。当人们想要研究相同群的不可约模表示时，也会出现量子化的包络代数(或量子群)。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在Hecke代数上以及M.Kashiwara和G.Lusztig在量子化包络代数上发现的正则基来研究。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，我们将研究量子化包络代数和Hecke代数的这些标准基和胞格表示。