Representation Theory, Cluster algebras, and Canonical Bases

PI计划在李代数和量子群的表示论，簇代数和非交换代数几何的接口的数学领域进行研究。 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。 它对数学的其他领域以及物理学产生了巨大的影响。 这个提议的一个中心主题是理解和研究正则基和晶体基的性质，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，该结构已被证明对于理解这些代数及其作为矩阵的实现是富有成效的。 理解这些基和紧密相关的结构之间的关系，如全正簇、几何晶体和簇代数，以及它们的量子和完全非交换类似物，是这个提议的一个统一主题。在一个项目中，PI提出了一种新的方法来研究Hecke代数，基于一些新的Hopf代数的发现：Hecke-Hopf代数包含了Hecke代数，同时又共同作用于它们。 从这项研究中得到的新信息允许量子杨-巴克斯特方程的新的解决方案的建设。 另一个项目的建议是探索新的基地，在量化包络代数具有显着的性质，如辫子群作用和兼容性与约瑟夫分解的量化包络代数。 这些新的基有望解决表示的自同态代数的分解问题，并帮助显式计算环境代数的中心。从这项研究中得到的新信息将被应用于构建新的量子和非交换簇结构，并证明量子Gelfand-Kirillov猜想为这些代数。

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.