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.

该研究提案属于算子代数领域，可以看作是对C*代数和冯诺依曼代数的研究。这两种代数都是在 20 世纪 20 年代末和 30 年代初引入的。 该提案部分致力于研究与相反代数非同构的 C* 代数的一些开放问题。拟议研究的基本原理是，要构建具有某些所需属性的 C* 代数示例，可以在冯诺依曼代数设置而不是 C* 代数设置中工作。我打算研究的具体问题涉及在 C* 代数理论中发挥重要作用的概念，即核性和纯无限 C* 代数。通过这个提案，我打算通过提供以前未知的 C* 代数示例，为 Elliott 的 C* 代数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群和量子排列群相关的开放问题。自由量子群是最近的研究领域，并且仍然是相当神秘的物体。这项工作的目的是更好地理解与自由量子群相关的冯诺依曼代数的性质和结构，因为自由量子群及其冯诺依曼代数在数学物理中具有重要的应用。最后，我计划用子因子理论证明代数组合学中的一个猜想。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一定作用下不变量的函数。这一作用导致了 Temperley-Lieb 代数在多项式环上的忠实作用。该猜想声称，Temperley-Lieb 代数的作用对 Fuss-Catalan 代数的忠实作用存在独特的扩展。解决这个猜想将成为研究在富斯-加泰罗尼亚代数作用下不变的函数的起点。希望这项研究能够像拟对称函数的研究一样有趣且成果丰富。