Representation Theory, Cluster algebras, and Canonical Bases

PI计划在李代数和量子群、簇代数和非对易代数几何的表示理论的交界处进行数学领域的研究。李代数和量子群的表示论是现代数学中一个动态发展的领域。它在数学的其他领域产生了巨大的影响，在物理学中也是如此。这个提议的一个中心主题是理解和研究标准基和结晶基的性质，它们赋予某些基本代数(如量子化包络代数)一个额外的结构，这种结构已被证明对于理解这些代数及其作为矩阵的实现是卓有成效的。在一个项目中，PI基于一些新的Hopf代数的发现，提出了一种新的研究Hecke代数的方法：包含Hecke代数并同时与其余作用的Hecke-Hopf代数。从这项研究中得到的新信息使得构造量子杨-巴克斯特方程的新解成为可能。该方案的另一个项目是探索量子化包络代数中具有辫子群作用和与量化包络代数的Joseph分解相容等显着性质的新的基。这些新的基可望解决表示的自同态代数的分解问题，并有助于显式地计算环境代数的中心。这项研究的新信息将被用于构造新的量子和非对易团簇结构，并用于证明这些代数的量子Gelfand-Kirillov猜想。

The PI plans to conduct research in an area of mathematics at the interface of the representation theory of Lie algebras and quantum groups, cluster algebras, and noncommutative algebraic geometry. The representation theory of Lie algebras and quantum groups is a dynamically developing field of modern mathematics. It has had a large impact in other areas of mathematics, as well as in physics. A central theme of this proposal is to understand and investigate the properties of canonical and crystal bases, which endow certain fundamental algebras (such as quantized enveloping algebras) with an additional structure that has proven to be fruitful for understanding these algebras and their realizations as matrices. Understanding the relationship between these bases and closely related structures, such as totally positive varieties, geometric crystals, and cluster algebras, and their quantum and totally noncommutative analogues, is a unifying theme of this proposal.In one project the PI proposes a new approach to the study of Hecke algebras based on the discovery of some new Hopf algebras: the Hecke-Hopf algebras which contain Hecke algebras and co-act on them at the same time. The new information resulting from this study allows for the construction of new solutions to the quantum Yang-Baxter equation. Another project in the proposal is to explore new bases in quantized enveloping algebras which possess remarkable properties such as braid group action and the compatibility with Joseph's decomposition of the quantized enveloping algebras. These new bases are expected to settle the problem of decomposing endomorphism algebras of representations and to help explicitly compute the center of the ambient algebra. New information resulting from this study will be applied to constructing new quantum and noncommutative cluster structures and to the proof of the quantum Gelfand-Kirillov conjecture for those algebras.