Approaching some open problems in algebraic combinatorics and in C*-algebras theory using von Neumann algebras.

使用冯诺依曼代数解决代数组合学和 C* 代数理论中的一些开放问题。

• 批准号: 386687-2012
• 负责人: Viola, MariaGrazia
• 金额: $ 1.09万
• 依托单位: Lakehead University
• 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=573012
• 关键词: Approaching; some; open; problems; algebraic

The research proposal is in the area of Operator Algebras, which can be viewed as the study of C*-algebras and von Neumann algebras. Both kinds of algebras were introduced between the late '20s and the early '30s. The proposal is partly devoted to the investigation of some open questions on C*-algebras non-isomorphic to their opposite algebra. The underlying philosophy of the proposed research is that to construct an example of a C*-algebra with some desired property, one can work in a von Neumann algebra setting instead of a C*-algebra setting. The specific problems that I intend to investigate deal with concepts that play a major role in the theory of C*-algebras, namely nuclearity and purely infinite C*-algebras. With this proposal I intend to provide some insights into Elliott's classification program for C*-algebras, by producing examples of C*-algebras which were not known before. In addition, I plan to investigate some open problems related to free quantum orthogonal groups and quantum permutation groups. Free quantum group are a recent area of research, and are still quite mysterious objects. The objective of this work is to gain a better understanding of the properties and structure of the von Neumann algebras associated to free quantume groups, since free quantum groups and their von Neumann algebras have important applications to mathematical physics. Lastly, I plan to prove a conjecture in algebraic combinatorics using subfactor theory. There has been an increased interest in recent years in quasi-symmetric functions. Quasi-symmetric functions are functions which are invariants under a certain action of the symmetric group on the ring of polynomials. This action induces a faithful action of the Temperley- Lieb algebra on the ring of polynomials. The conjecture claims that there is a unique extensions of the action of the Temperley-Lieb algebra to a faithful action of the Fuss-Catalan algebra. Solving this conjecture will be the starting point of the study of functions which are invariant under the action of the Fuss-Catalan algebra. Hopefully this study will be as interesting and as rich in results as the study of quasi-symmetric functions has been.

研究建议是在操作员代数领域的，可以将其视为对C*-Elgebras和von Neumann代数的研究。在20年代末至30年代初期，引入了两种代数。 该提案部分致力于调查有关其相反代数的C* - 代数非同构的一些开放问题。拟议的研究的基本哲学是，要构建具有一些理想特性的C*代数的示例，可以在von Neumann代数设置中工作，而不是C*-Algebra设置。我打算调查的具体问题处理在C* - 代数理论中起主要作用的概念，即核性和纯粹的无限C* - 代数。通过此提案，我打算通过制作以前未知的C*-代代代数的示例来对Elliott的C*-Algebras的分类程序提供一些见解。此外，我计划研究一些与自由量子正交组和量子置换组有关的开放问题。自由量子组是最近的研究领域，仍然是非常神秘的对象。这项工作的目的是更好地了解与自由量子组相关的von Neumann代数的性质和结构，因为自由量子组及其von Neumann代数对数学物理学具有重要的应用。最后，我计划使用子因子理论证明代数组合学中的猜想。近年来，对准对称功能的兴趣增加了。准对称函数是在对称组对多项式环的某个作用下不变的函数。这种行动引起了temperley-lieb代数在多项式环上的忠实行动。该猜想声称，temperley-lieb代数的行动具有独特的扩展，以实现大惊小怪的代数的忠实行动。解决这一猜想将是研究功能研究的起点，这些功能是在大惊小怪的代数的作用下不变的。希望这项研究将像准对称功能的研究一样有趣且丰富的结果。