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: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635293
• 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.

研究方向为算子代数，可视为C*代数和冯·诺伊曼代数的研究。这两种代数都是在20年代末到30年代初被引入的。本文部分致力于对C*-代数非同构的若干开放性问题的研究。所提出的研究的基本哲学是，为了构造一个具有某些期望性质的C*-代数的例子，可以在冯·诺伊曼代数设置中工作，而不是C*-代数设置。我打算研究的具体问题涉及在C*-代数理论中起主要作用的概念，即核和纯无限C*-代数。有了这个建议，我打算通过产生以前不知道的C*代数的例子，来提供一些关于艾略特C*代数分类程序的见解。此外，我计划研究一些与自由量子正交群和量子置换群有关的开放问题。自由量子群是近年来的一个研究领域，仍然是一个相当神秘的物体。这项工作的目的是为了更好地理解与自由量子群相关的冯·诺伊曼代数的性质和结构，因为自由量子群及其冯·诺伊曼代数在数学物理中有重要的应用。最后，我打算用子因子理论证明代数组合中的一个猜想。近年来，人们对拟对称函数越来越感兴趣。拟对称函数是在多项式环上对称群的一定作用下不变量的函数。这个作用引起了多项式环上的坦波利-利布代数的忠实作用。这个猜想声称坦波利-利布代数的作用对富斯-卡塔兰代数的忠实作用有一个唯一的延伸。这一猜想的解决将成为研究在Fuss-Catalan代数作用下不变量的函数的出发点。希望这项研究能像准对称函数的研究一样有趣，结果丰富。